Harmonic series: Difference between revisions

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4,550th iteration of the harmonic series, the sum is greater than 9
4,550th iteration of the harmonic series, the sum is greater than 9
12,367th iteration of the harmonic series, the sum is greater than 10
12,367th iteration of the harmonic series, the sum is greater than 10
</pre>

=={{header|Ring}}==
<lang ring>
decimals(12)
sum = 0
nNew = 1
limit = 13000
Harmonic = []


for n = 1 to limit
sum += 1/n
add(Harmonic,[n,sum])
next

see "The first twenty harmonic numbers are:" + nl
for n = 1 to 20
see "" + Harmonic[n][1] + " -> " + Harmonic[n][2] + nl
next
see nl

for m = 1 to 10
for n = nNew to len(Harmonic)
if Harmonic[n][2] > m
see "The first harmonic number greater than "
see "" + m + " is " + Harmonic[n][2] + ", at position " + n + nl
nNew = n
exit
ok
next
next
</lang>
{{out}}
<pre>
The first twenty harmonic numbers are:
1 -> 1
2 -> 1.500000000000
3 -> 1.833333333333
4 -> 2.083333333333
5 -> 2.283333333333
6 -> 2.450000000000
7 -> 2.592857142857
8 -> 2.717857142857
9 -> 2.828968253968
10 -> 2.928968253968
11 -> 3.019877344877
12 -> 3.103210678211
13 -> 3.180133755134
14 -> 3.251562326562
15 -> 3.318228993229
16 -> 3.380728993229
17 -> 3.439552522641
18 -> 3.495108078196
19 -> 3.547739657144
20 -> 3.597739657144

The first harmonic number greater than 1 is 1.500000000000, at position 2
The first harmonic number greater than 2 is 2.083333333333, at position 4
The first harmonic number greater than 3 is 3.019877344877, at position 11
The first harmonic number greater than 4 is 4.027245195437, at position 31
The first harmonic number greater than 5 is 5.002068272680, at position 83
The first harmonic number greater than 6 is 6.004366708346, at position 227
The first harmonic number greater than 7 is 7.001274097134, at position 616
The first harmonic number greater than 8 is 8.000485571996, at position 1674
The first harmonic number greater than 9 is 9.000208062931, at position 4550
The first harmonic number greater than 10 is 10.000043008276, at position 12367
</pre>
</pre>


Revision as of 13:28, 10 January 2022

Harmonic series is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
This page uses content from Wikipedia. The original article was at Harmonic number. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)


In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: 

   Hn = 1 + 1/2 + 1/3 + ... + 1/n

The series of harmonic numbers thus obtained is often loosely referred to as the harmonic series.

Harmonic numbers are closely related to the Riemann zeta function, and roughly approximate the natural logarithm function; differing by γ (lowercase Gamma), the Euler–Mascheroni constant.

The harmonic series is divergent, albeit quite slowly, and grows toward infinity.


Task
  • Write a function (routine, procedure, whatever it may be called in your language) to generate harmonic numbers.
  • Use that procedure to show the values of the first 20 harmonic numbers.
  • Find and show the position in the series of the first value greater than the integers 1 through 5


Stretch
  • Find and show the position in the series of the first value greater than the integers 6 through 10


Related


ALGOL 68

Using standard lenghth REAL numbers this can find the first Harmonic number > 10, would probably need higher precision to find Harmonic numbers with larger values. <lang algol68>BEGIN # find some harmonic numbers, Hn is the sum if the reciprocals of 1..n # 

   # returns the first n Harmonic numbers #
   OP   HARMONIC = ( INT n )[]REAL:
        BEGIN
           [ 1 : n ]REAL h;
           h[ 1 ] := 1;
           FOR i FROM 2 TO n DO
               h[ i ] := h[ i - 1 ] + ( 1 / i )
           OD;
           h
        END # HARMONIC # ;
   # find the first 20 000 harmonic numbers #
   []REAL h = HARMONIC 20 000;
   # show the first 20 harmonic numbers #
   FOR i TO 20 DO
       print( ( whole( i, -2 ), ":", fixed( h[ i ], -14, 8 ), newline ) )
   OD;
   # find the positions of the first harmonic number > n where n in 1... #
   INT  rqd int  := 1;
   REAL rqd real := 1;
   FOR i TO UPB h DO
       IF h[ i ] > rqd real THEN
           # found the first harmonic number greater than rqd real #
           print( ( "Position of the first harmonic number > ", whole( rqd int, -2 ), ": ", whole( i, 0 ), newline ) );
           rqd int  +:= 1;
           rqd real +:= 1
       FI
   OD

END</lang>

Output:
 1:    1.00000000
 2:    1.50000000
 3:    1.83333333
 4:    2.08333333
 5:    2.28333333
 6:    2.45000000
 7:    2.59285714
 8:    2.71785714
 9:    2.82896825
10:    2.92896825
11:    3.01987734
12:    3.10321068
13:    3.18013376
14:    3.25156233
15:    3.31822899
16:    3.38072899
17:    3.43955252
18:    3.49510808
19:    3.54773966
20:    3.59773966
Position of the first harmonic number >  1: 2
Position of the first harmonic number >  2: 4
Position of the first harmonic number >  3: 11
Position of the first harmonic number >  4: 31
Position of the first harmonic number >  5: 83
Position of the first harmonic number >  6: 227
Position of the first harmonic number >  7: 616
Position of the first harmonic number >  8: 1674
Position of the first harmonic number >  9: 4550
Position of the first harmonic number > 10: 12367

AWK

<lang AWK>

  1. syntax: GAWK -f HARMONIC_SERIES.AWK
  2. converted from FreeBASIC

BEGIN { 

   limit = 20
   printf("The first %d harmonic numbers:\n",limit)
   for (n=1; n<=limit; n++) {
     h += 1/n
     printf("%2d %11.8f\n",n,h)
   }
   print("")
   h = 1
   n = 2
   for (i=2; i<=10; i++) {
     while (h < i) {
       h += 1/n
       n++
     }
     printf("The first harmonic number > %2d is %11.8f at position %d\n",i,h,n-1)
   }
   exit(0)

} </lang>

Output:
The first 20 harmonic numbers:
 1  1.00000000
 2  1.50000000
 3  1.83333333
 4  2.08333333
 5  2.28333333
 6  2.45000000
 7  2.59285714
 8  2.71785714
 9  2.82896825
10  2.92896825
11  3.01987734
12  3.10321068
13  3.18013376
14  3.25156233
15  3.31822899
16  3.38072899
17  3.43955252
18  3.49510808
19  3.54773966
20  3.59773966

The first harmonic number >  2 is  2.08333333 at position 4
The first harmonic number >  3 is  3.01987734 at position 11
The first harmonic number >  4 is  4.02724520 at position 31
The first harmonic number >  5 is  5.00206827 at position 83
The first harmonic number >  6 is  6.00436671 at position 227
The first harmonic number >  7 is  7.00127410 at position 616
The first harmonic number >  8 is  8.00048557 at position 1674
The first harmonic number >  9 is  9.00020806 at position 4550
The first harmonic number > 10 is 10.00004301 at position 12367

C++

Library: Boost

<lang cpp>#include <iomanip>

  1. include <iostream>
  2. include <boost/rational.hpp>
  3. include <boost/multiprecision/gmp.hpp>

using integer = boost::multiprecision::mpz_int; using rational = boost::rational<integer>;

class harmonic_generator { public: 

   rational next() {
       rational result = term_;
       term_ += rational(1, ++n_);
       return result;
   }
   void reset() {
       n_ = 1;
       term_ = 1;
   }

private: 

   integer n_ = 1;
   rational term_ = 1;

};

int main() { 

   std::cout << "First 20 harmonic numbers:\n";
   harmonic_generator hgen;
   for (int i = 1; i <= 20; ++i)
       std::cout << std::setw(2) << i << ". " << hgen.next() << '\n';
   
   rational h;
   for (int i = 1; i <= 80; ++i)
       h = hgen.next();
   std::cout << "\n100th harmonic number: " << h << "\n\n";

   int n = 1;
   hgen.reset();
   for (int i = 1; n <= 10; ++i) {
       if (hgen.next() > n)
           std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n';
   }

}</lang>

Output:
First 20 harmonic numbers:
 1. 1/1
 2. 3/2
 3. 11/6
 4. 25/12
 5. 137/60
 6. 49/20
 7. 363/140
 8. 761/280
 9. 7129/2520
10. 7381/2520
11. 83711/27720
12. 86021/27720
13. 1145993/360360
14. 1171733/360360
15. 1195757/360360
16. 2436559/720720
17. 42142223/12252240
18. 14274301/4084080
19. 275295799/77597520
20. 55835135/15519504

100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

Position of first term >  1: 2
Position of first term >  2: 4
Position of first term >  3: 11
Position of first term >  4: 31
Position of first term >  5: 83
Position of first term >  6: 227
Position of first term >  7: 616
Position of first term >  8: 1674
Position of first term >  9: 4550
Position of first term > 10: 12367

COBOL

<lang cobol> IDENTIFICATION DIVISION. 

      PROGRAM-ID. HARMONIC.
      
      DATA DIVISION.
      WORKING-STORAGE SECTION.
      01 VARS.
         03 N           PIC 9(5) VALUE ZERO.
         03 HN          PIC 9(2)V9(12) VALUE ZERO.
         03 INT         PIC 99 VALUE ZERO.
      01 OUT-VARS.
         03 POS         PIC Z(4)9.
         03 FILLER      PIC X(3) VALUE SPACES.
         03 H-OUT       PIC Z9.9(12).

      PROCEDURE DIVISION.
      BEGIN.
          DISPLAY "First 20 harmonic numbers:"
          PERFORM SHOW-HARMONIC 20 TIMES.
          DISPLAY SPACES.
          MOVE ZERO TO N, HN.
          DISPLAY "First harmonic number to exceed whole number:"
          PERFORM EXCEED-INT 10 TIMES.
          STOP RUN.

      SHOW-HARMONIC.
          PERFORM NEXT-HARMONIC.
          MOVE HN TO H-OUT.
          DISPLAY H-OUT.

      EXCEED-INT.
          ADD 1 TO INT.
          PERFORM NEXT-HARMONIC UNTIL HN IS GREATER THAN INT.
          MOVE N TO POS.
          MOVE HN TO H-OUT.
          DISPLAY OUT-VARS.

      NEXT-HARMONIC.
          ADD 1 TO N.
          COMPUTE HN = HN + 1 / N.</lang>
Output:
First 20 harmonic numbers:
 1.000000000000
 1.500000000000
 1.833333333333
 2.083333333333
 2.283333333333
 2.449999999999
 2.592857142856
 2.717857142856
 2.828968253967
 2.928968253967
 3.019877344876
 3.103210678209
 3.180133755132
 3.251562326560
 3.318228993226
 3.380728993226
 3.439552522637
 3.495108078192
 3.547739657139
 3.597739657139
 
First harmonic number to exceed whole number:
    2    1.500000000000
    4    2.083333333333
   11    3.019877344876
   31    4.027245195428
   83    5.002068272651
  227    6.004366708257
  616    7.001274096877
 1674    8.000485571261
 4550    9.000208060802
12367   10.000043002313

Factor

This solution uses the following (rather accurate) approximation of the harmonic numbers to find the first indices greater than the integers:

Hn ≈ ln(n) + γ + 1/2n - 1/12n2

where γ is the Euler-Mascheroni constant, approximately 0.5772156649.

Works with: Factor version 0.99 2021-02-05

<lang factor>USING: formatting grouping io kernel lists lists.lazy math math.functions math.ranges math.statistics math.text.english prettyprint sequences tools.memory.private ;

! Euler-Mascheroni constant CONSTANT: γ 0.5772156649

Hn-approx ( n -- ~Hn )
   [ log γ + 1 2 ] [ * /f + 1 ] [ sq 12 * /f - ] tri ;
lharmonics ( -- list ) 1 lfrom [ Hn-approx ] lmap-lazy ;
first-gt ( m -- n ) lharmonics swap '[ _ < ] lwhile llength ;

"First twenty harmonic numbers as mixed numbers:" print 100 [1,b] [ recip ] map cum-sum [ 20 head 5 group simple-table. nl ] [ "One hundredth:" print last . nl ] bi

"(zero based) Index of first value:" print 10 [1,b] [ 

   dup first-gt [ commas ] [ 1 + number>text ] bi
   "  greater than %2d: %6s (term number %s)\n" printf

] each</lang>

Output:
First twenty harmonic numbers as mixed numbers:
1               1+1/2              1+5/6             2+1/12              2+17/60
2+9/20          2+83/140           2+201/280         2+2089/2520         2+2341/2520
3+551/27720     3+2861/27720       3+64913/360360    3+90653/360360      3+114677/360360
3+274399/720720 3+5385503/12252240 3+2022061/4084080 3+42503239/77597520 3+9276623/15519504

One hundredth:
5+522561233577855727314756256041670736351/2788815009188499086581352357412492142272

(zero based) Index of first value:
  greater than  1:      1 (term number two)
  greater than  2:      3 (term number four)
  greater than  3:     10 (term number eleven)
  greater than  4:     30 (term number thirty-one)
  greater than  5:     82 (term number eighty-three)
  greater than  6:    226 (term number two hundred and twenty-seven)
  greater than  7:    615 (term number six hundred and sixteen)
  greater than  8:  1,673 (term number one thousand, six hundred and seventy-four)
  greater than  9:  4,549 (term number four thousand, five hundred and fifty)
  greater than 10: 12,366 (term number twelve thousand, three hundred and sixty-seven)

Forth

Uses fixed point computation which is more traditional in FORTH. <lang Forth> warnings off

1.000.000.000.000.000 drop constant 1.0fx \ fractional part is 15 decimal digits.

.h ( n -- )
   s>d <#  14 for # next  [char] . hold #s #> type space ;

1.0fx 1 2constant first-harmonic

round 5 + 10 / ;
next-harmonic ( h n -- h' n' )
   1+ tuck [ 1.0fx 10 * ] literal swap / round + swap ;
task1
   first-harmonic  19 for  over cr .h next-harmonic  next 2drop ;
task2
   first-harmonic
   11 1 do
       begin over i 1.0fx * <= while
           next-harmonic
       repeat
       dup .
   loop 2drop ;

." The first 10 harmonic numbers: " task1 cr cr ." The nth index of the first harmonic number that exceeds the nth integer: " cr task2 cr bye </lang>

Output:
The first 10 harmonic numbers: 
1.000000000000000 
1.500000000000000 
1.833333333333333 
2.083333333333333 
2.283333333333333 
2.450000000000000 
2.592857142857143 
2.717857142857143 
2.828968253968254 
2.928968253968254 
3.019877344877345 
3.103210678210678 
3.180133755133755 
3.251562326562326 
3.318228993228993 
3.380728993228993 
3.439552522640758 
3.495108078196314 
3.547739657143682 
3.597739657143682 

The nth index of the first harmonic number that exceeds the nth integer: 
2 4 11 31 83 227 616 1674 4550 12367

FreeBASIC

<lang freebasic>dim as double h = 0.0 dim as uinteger n, i

print "The first twenty harmonic numbers are:" for n = 1 to 20 

   h += 1.0/n
   print n, h

next n

h = 1 : n = 2 for i=2 to 10 

   while h<i
       h+=1.0/n
       n+=1
   wend
   print "The first harmonic number greater than ";i;" is ";h;", at position ";n-1

next i</lang>

Output:
The first twenty harmonic numbers are:

1              1
2              1.5
3              1.833333333333333
4              2.083333333333333
5              2.283333333333333
6              2.45
7              2.592857142857143
8              2.717857142857143
9              2.828968253968254
10             2.928968253968254
11             3.019877344877345
12             3.103210678210678
13             3.180133755133755
14             3.251562326562327
15             3.318228993228994
16             3.380728993228994
17             3.439552522640758
18             3.495108078196314
19             3.547739657143682
20             3.597739657143682
The first harmonic number greater than 2 is  2.083333333333333, at position 4
The first harmonic number greater than 3 is  3.019877344877345, at position 11
The first harmonic number greater than 4 is  4.02724519543652, at position 31
The first harmonic number greater than 5 is  5.002068272680166, at position 83
The first harmonic number greater than 6 is  6.004366708345567, at position 227
The first harmonic number greater than 7 is  7.001274097134162, at position 616
The first harmonic number greater than 8 is  8.000485571995782, at position 1674
The first harmonic number greater than 9 is  9.000208062931115, at position 4550


The first harmonic number greater than 10 is  10.00004300827578, at position 12367

Go

Translation of: Wren

<lang go>package main

import ( 

   "fmt"
   "math/big"

)

func harmonic(n int) *big.Rat { 

   sum := new(big.Rat)
   for i := int64(1); i <= int64(n); i++ {
       r := big.NewRat(1, i)
       sum.Add(sum, r)
   }
   return sum

}

func main() { 

   fmt.Println("The first 20 harmonic numbers and the 100th, expressed in rational form, are:")
   numbers := make([]int, 21)
   for i := 1; i <= 20; i++ {
       numbers[i-1] = i
   }
   numbers[20] = 100
   for _, i := range numbers {
       fmt.Printf("%3d : %s\n", i, harmonic(i))
   }

   fmt.Println("\nThe first harmonic number to exceed the following integers is:")
   const limit = 10
   for i, n, h := 1, 1, 0.0; i <= limit; n++ {
       h += 1.0 / float64(n)
       if h > float64(i) {
           fmt.Printf("integer = %2d  -> n = %6d  ->  harmonic number = %9.6f (to 6dp)\n", i, n, h)
           i++
       }
   }

}</lang>

Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are:
  1 : 1/1
  2 : 3/2
  3 : 11/6
  4 : 25/12
  5 : 137/60
  6 : 49/20
  7 : 363/140
  8 : 761/280
  9 : 7129/2520
 10 : 7381/2520
 11 : 83711/27720
 12 : 86021/27720
 13 : 1145993/360360
 14 : 1171733/360360
 15 : 1195757/360360
 16 : 2436559/720720
 17 : 42142223/12252240
 18 : 14274301/4084080
 19 : 275295799/77597520
 20 : 55835135/15519504
100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

The first harmonic number to exceed the following integers is:
integer =  1  -> n =      2  ->  harmonic number =  1.500000 (to 6dp)
integer =  2  -> n =      4  ->  harmonic number =  2.083333 (to 6dp)
integer =  3  -> n =     11  ->  harmonic number =  3.019877 (to 6dp)
integer =  4  -> n =     31  ->  harmonic number =  4.027245 (to 6dp)
integer =  5  -> n =     83  ->  harmonic number =  5.002068 (to 6dp)
integer =  6  -> n =    227  ->  harmonic number =  6.004367 (to 6dp)
integer =  7  -> n =    616  ->  harmonic number =  7.001274 (to 6dp)
integer =  8  -> n =   1674  ->  harmonic number =  8.000486 (to 6dp)
integer =  9  -> n =   4550  ->  harmonic number =  9.000208 (to 6dp)
integer = 10  -> n =  12367  ->  harmonic number = 10.000043 (to 6dp)

Haskell

<lang haskell>import Data.List (find) import Data.Ratio

HARMONIC SERIES --------------------

harmonic :: [Rational] harmonic = 

 scanl1
   (\a x -> a + 1 / x)
   [1 ..]

TESTS -------------------------

main :: IO () main = do 

 putStrLn "First 20 terms:"
 mapM_ putStrLn $
   showRatio <$> take 20 harmonic

 putStrLn "\n100th term:"
 putStrLn $ showRatio (harmonic !! 99)
 putStrLn ""

 putStrLn "One-based indices of first terms above threshold values:"
 let indexedHarmonic = zip [0 ..] harmonic
 mapM_
   putStrLn
   $ fmap
     ( showFirstLimit
         <*> \n -> find ((> n) . snd) indexedHarmonic
     )
     [1 .. 10]

DISPLAY FORMATTING ------------------

showFirstLimit n (Just (i, r)) = 

 "Term "
   <> show (succ i)
   <> " is the first above "
   <> show (numerator n)

showRatio :: Ratio Integer -> String showRatio = 

 ((<>) . show . numerator)
   <*> (('/' :) . show . denominator)</lang>
Output:
First 20 terms:
1/1
3/2
11/6
25/12
137/60
49/20
363/140
761/280
7129/2520
7381/2520
83711/27720
86021/27720
1145993/360360
1171733/360360
1195757/360360
2436559/720720
42142223/12252240
14274301/4084080
275295799/77597520
55835135/15519504

100th term:
14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

One-based indices of first terms above threshold values:
Term 2 is the first above 1
Term 4 is the first above 2
Term 11 is the first above 3
Term 31 is the first above 4
Term 83 is the first above 5
Term 227 is the first above 6
Term 616 is the first above 7
Term 1674 is the first above 8
Term 4550 is the first above 9
Term 12367 is the first above 10

jq

Translation of: Wren

Works with gojq, the Go implementation of jq

This entry requires a rational arithmetic module such as is available at Arithmetic/Rational#jq. <lang jq># include "rational"; # a reminder

def harmonic: 

 reduce range(1; 1+.) as $i ( r(0;1);
   radd(.; r(1; $i) ));

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;

def task1: 

 "The first 20 harmonic numbers and the 100th, expressed in rational form, are:",
 (range(1;21), 100
  | "\(.) : \(harmonic|rpp)" );

def task2($limit): 

 "The first harmonic number to exceed the following integers is:",
     limit($limit;
   foreach range(0; infinite) as $n (
     {i: 1, n: 1, h: r(0;1)};
     .emit = false
     | .h = radd(.h; r(1; .n))
     | .i as $i
     | if .h | rgreaterthan($i)
       then .emit = "integer = \(.i|lpad(2))  -> n = \(.n| lpad(6))  ->  harmonic number = \(.h|r_to_decimal(6)) (to 6dp)"

| .i += 1 

       else .
       end 
       | .n += 1;
     select(.emit).emit) );

task1, "", task2(10)</lang>

Output:

The first 20 harmonic numbers and the 100th, expressed in rational form, are:
1 : 1 // 1
2 : 3 // 2
3 : 11 // 6
4 : 25 // 12
5 : 137 // 60
6 : 49 // 20
7 : 363 // 140
8 : 761 // 280
9 : 7129 // 2520
10 : 7381 // 2520
11 : 83711 // 27720
12 : 86021 // 27720
13 : 1145993 // 360360
14 : 1171733 // 360360
15 : 1195757 // 360360
16 : 2436559 // 720720
17 : 42142223 // 12252240
18 : 14274301 // 4084080
19 : 275295799 // 77597520
20 : 55835135 // 15519504
100 : 14466636279520351160221518043104131447711 // 2788815009188499086581352357412492142272

The first harmonic number to exceed the following integers is:
integer =  1  -> n =      2  ->  harmonic number = 1.5 (to 6dp)
integer =  2  -> n =      4  ->  harmonic number = 2.083333 (to 6dp)
integer =  3  -> n =     11  ->  harmonic number = 3.019877 (to 6dp)
integer =  4  -> n =     31  ->  harmonic number = 4.027245 (to 6dp)
integer =  5  -> n =     83  ->  harmonic number = 5.002068 (to 6dp)
integer =  6  -> n =    227  ->  harmonic number = 6.004366 (to 6dp)
integer =  7  -> n =    616  ->  harmonic number = 7.001274 (to 6dp)
integer =  8  -> n =   1674  ->  harmonic number = 8.000485 (to 6dp)
integer =  9  -> n =   4550  ->  harmonic number = 9.000208 (to 6dp)
integer = 10  -> n =  12367  ->  harmonic number = 10.000043 (to 6dp)


Julia

<lang julia>const memoizer = [BigFloat(1.0), BigFloat(1.5)]

""" 

   harmonic(n::Integer)::BigFloat

Calculates harmonic numbers. The integer argument `n` should be positive. """ function harmonic(n::Integer)::BigFloat 

   if n < 0
       throw(DomainError(n))
   elseif n == 0
       return BigFloat(0.0)   # by convention
   elseif length(memoizer) >= n
       return memoizer[n]
   elseif length(memoizer) + 1 == n
       h = memoizer[end] + BigFloat(1.0) / n
       push!(memoizer, h)
       return h
   elseif n < 1_000_000
       start, x = length(memoizer), memoizer[end]
       for i in start+1:n
           push!(memoizer, (x += big"1.0" / i))
       end
       return memoizer[end]
   else
       # use H(n) = eulergamma + digamma(n + 1), instead, if memory use of memoization too large
       x = n + big"1.0"
       digam = BigFloat()
       ccall((:mpfr_digamma, :libmpfr), Int32, (Ref{BigFloat}, Ref{BigFloat}, Int32), digam, x, 1)
       return Base.MathConstants.eulergamma + digam
   end

end

function testharmonics(upperlimit = 11) 

   n = 1
   while (h = harmonic(n)) < upperlimit
       nextintegerfloor = h < 1.8 ? h > 1.0 : floor(h) > floor(memoizer[n - 1])
       if n < 21 || nextintegerfloor
           println("harmonic($n) = $h")
           nextintegerfloor && println("    $n is also the term number for the first harmonic > $(floor(h))")
       end
       n += 1
   end

end

testharmonics()

</lang>

Output:
harmonic(1) = 1.0
harmonic(2) = 1.5
    2 is also the term number for the first harmonic > 1.0
harmonic(3) = 1.833333333333333333333333333333333333333333333333333333333333333333333333333339
harmonic(4) = 2.083333333333333333333333333333333333333333333333333333333333333333333333333356
    4 is also the term number for the first harmonic > 2.0
harmonic(5) = 2.283333333333333333333333333333333333333333333333333333333333333333333333333363
harmonic(6) = 2.450000000000000000000000000000000000000000000000000000000000000000000000000041
harmonic(7) = 2.592857142857142857142857142857142857142857142857142857142857142857142857142913
harmonic(8) = 2.717857142857142857142857142857142857142857142857142857142857142857142857142913
harmonic(9) = 2.828968253968253968253968253968253968253968253968253968253968253968253968254009
harmonic(10) = 2.928968253968253968253968253968253968253968253968253968253968253968253968253995
harmonic(11) = 3.019877344877344877344877344877344877344877344877344877344877344877344877344889
    11 is also the term number for the first harmonic > 3.0
harmonic(12) = 3.103210678210678210678210678210678210678210678210678210678210678210678210678211
harmonic(13) = 3.180133755133755133755133755133755133755133755133755133755133755133755133755123
harmonic(14) = 3.251562326562326562326562326562326562326562326562326562326562326562326562326542
harmonic(15) = 3.318228993228993228993228993228993228993228993228993228993228993228993228993199
harmonic(16) = 3.380728993228993228993228993228993228993228993228993228993228993228993228993199
harmonic(17) = 3.439552522640757934875581934405463817228523110876052052522640757934875581934384
harmonic(18) = 3.495108078196313490431137489961019372784078666431607608078196313490431137489932
harmonic(19) = 3.547739657143681911483769068908387793836710245378976029130827892437799558542556
harmonic(20) = 3.597739657143681911483769068908387793836710245378976029130827892437799558542549
harmonic(31) = 4.027245195436520102759838180253409570739320924649712368107240380481568735938418
    31 is also the term number for the first harmonic > 4.0
harmonic(83) = 5.002068272680166053728324750753870264345455215566438587478989543061001039767003
    83 is also the term number for the first harmonic > 5.0
harmonic(227) = 6.004366708345566023376436217157408474650893771305512336984772241757969069086895
    227 is also the term number for the first harmonic > 6.0
harmonic(616) = 7.001274097134160381487068933022945074864048309674852535721112060499845844673362
    616 is also the term number for the first harmonic > 7.0
harmonic(1674) = 8.000485571995779067790304796519697445800341927883408389172647367923220595045883
    1674 is also the term number for the first harmonic > 8.0
harmonic(4550) = 9.000208062931140339164179501268928624268799275400095995932594006439583360896694
    4550 is also the term number for the first harmonic > 9.0
harmonic(12367) = 10.00004300827580769470675707492981720768686887243344211163998834649135547210551
    12367 is also the term number for the first harmonic > 10.0

Using rationals

<lang julia>const harmonics = accumulate((x, y) -> x + big"1" // y, 1:12370)

println("First twenty harmonic numbers as rationals:") foreach(i -> println(rpad(i, 3), " => ", harmonics[i]), 1:20)

println("\nThe 100th harmonic is: ", harmonics[100], "\n")

for n in 1:10 

   idx = findfirst(x -> x > n, harmonics)
   print("First Harmonic > $n is at position $idx and is: ", harmonics[idx], "\n\n")

end

</lang>

Output:
First twenty harmonic numbers as rationals:
1   => 1//1
2   => 3//2
3   => 11//6
4   => 25//12
5   => 137//60
6   => 49//20
7   => 363//140
8   => 761//280
9   => 7129//2520
10  => 7381//2520
11  => 83711//27720
12  => 86021//27720
13  => 1145993//360360
14  => 1171733//360360
15  => 1195757//360360
16  => 2436559//720720
17  => 42142223//12252240
18  => 14274301//4084080
19  => 275295799//77597520
20  => 55835135//15519504

The 100th harmonic is: 14466636279520351160221518043104131447711//2788815009188499086581352357412492142272

First Harmonic > 1 is at position 2 and is: 3//2

First Harmonic > 2 is at position 4 and is: 25//12

First Harmonic > 3 is at position 11 and is: 83711//27720

First Harmonic > 4 is at position 31 and is: 290774257297357//72201776446800

First Harmonic > 5 is at position 83 and is: 3672441655127796364812512959533039359//734184632222154704090370027645633600

First Harmonic > 6 is at position 227 and is: 7210530454341478178114292924106791866448071719960766673184657267908514585008387695857601640547547//1200881092808579751109445892858157237623011602251376919557525378451885327053551694768211209584000

First Harmonic > 7 is at position 616 and is: 32418148234584438506244669620559801586453637524407441287287956498408167324384836135154335608450103705554364675635095796463552565723833053990601655182066284053456674137000162471237344506126617371216079120095971345235097691744446098739930413505848076716148051590567//4630321250792651667959778962730272725135972962879403462498154236164728708558353376685441262094879300811505205446228182535680317471673227192574715015221837802633281044802046825192844588381756912000006314261374254853934981337617963082619967597058012665881508170240

First Harmonic > 8 is at position 1674 and is: 138141199730356031917751723608451630140926572041533266161867229808423134343694448776284861362361049658120307218628922472967074569643922410693236711646552594017799177544898923160715275467074773052672543691485042344006328346792717577331346270889819765648723902241194752204176829233330048155240810379245143159803746553605910654524991868894244655125224374793405027930983653080599361646248333997230466686371943554425849838679719632376766617639339476485319908534320267241696294676687857612211224929536683459738832437894009105247288571220395942120501422698013842452465990861198601652372525840473807472136998877762025722745653584203415416989419985647929410780044991971375219261681255528981314607252039408935696726675566989805504130221402879//17266601943998720215934555231097205020805360283925021776131771421603665580550992851730546192114435876915764081799263950036774496079790573533431946151195888729825227967898856935376506079740151630837994958679444177020560516804785312006925540853535681555388532126434276232761609235943363516882249874992501288848101663246418034482830782765799910587627015490435618612817485429045075697874945013812237906815356826216080083918931352849328226757321069450237292900402036556169613799106635542806391854137317272454464723030848621364955770185788658456980359320481573473968897297574022422610618929844739054324826111937140267616883649616842817036401014150739236179076410461198210703989272527503945999049527912447331310263532711648780174245760000

First Harmonic > 9 is at position 4550 and is: 2803922057204811543989535496612889221797101689401339733425779071893883726062730194418722759045899108563649222513327676111056373937430194191361767957531399947657882950860187200723323262653682200418079207248717851019834955117510807290247517344853824182604853734193915236890382944826941254750728989773414206859570065960640186792890664839103798433780558510849879082382309565191620282894846917562664482131202291813204472491830923673050493632880376511556753354676402403084278296415856939754434734260623967677761451430096042117604187063669249143393076890252523001247824132437003474131230135624887874799541971404156392012052221712958314528322355541629626096561834856317166367946519841625273109689443740339231545297896759513984852874078072318627511091395704617775835151474534768408675032414448963002972590468720049363107497714657975180640114587539853156735035634359134035816625004885128358508857789053399591546078553132392557669776329295754833648165907192091248381853554990310275294735796601315160268572473977239894244510958924644461475601889585298550864600912608515846981406115574447937712467306028124552530287697424685820096894634400504911945644842525080877704081133701601474836419212357427281193595545654309059483619700606698188346807113314394305087007324692154551457096864441413765832492015835459544797423019196304195496229963766867559041682555605485756372169731071238590755894190062578370776223859945908951232125318924762539326986977852885933136208484015119262988152215227210365689996539057011922584498047033647706625149283085817504520106280560129638082705378996661011957957116006944584486439354147266379272818775057457727700062964206158298239058238442033871955799108732891594547873095023995742714560400884479305635074935996707008596929664779547461570776674459690821497530879010144224813067523793975748941040890361283562073364309054362072137372458221297707667641071756676535258762295067395479041974831597965979613081474695278123265026663091006352059826017007472704096220374679625403//311539693038123496722215310551186904103082524726970170658802584453586817702845967012058559851648961565162714074357248615113237727615741409397613403942762027416418250197869235139807433555974792925305500675331577565501246819843747067123329275977858482555209853394601249731116273203314723492820718920663543363350452108859588055589207296367576316485295250028686463172112871115738305282269830733309445640796833467195761172540352742868324044527033042113461149657815168691435568328007659916683562048346389048741726140068665849102873300936534794943981012153540913055621237325128266901087332680399450826342551172911601227269655973249675723840414635853583858640961466968497444812002669481333031399627561658239662030094505016308687209615676717899935366285911202891060019766201048122485091665890488010192945266722188069264271709669076333959826410108104815827793429989042734394907438794428685359256959974813630940895555145059961227687918900885009633119015801758204737016204228694445315173976166116707848897795690334687813067638452214980721143416739719469158748498888549650173439285892765248219730073273036517725939409971809083058041780264560576764115967213013380206017646943405476909576091357482868476761698005324735669002986992261514164471647621973497050651105016702049275202360233254308086080840811811270395182950866854592166254651076234420662875446908943737749367221987863265181700223872816922018831416550743452655599236620182838774266078307310527709836462300005876748274266928980195184478251265986886488170211071786897470230189921260226806813948222307759474759342997757077589856537503856731410785456607398210129322637205537554981139121822428297380197652187006595903988741596680008103907700803965881407891948670316751093348441767470344198247057046756517630261990510817072323901911147038583224901683429072731095666462995891134319532658460042039832285713339493699639220237857309737109028479215274870712938102667472652582674547746273957596211401299308669708874527010828161490844069542400000

First Harmonic > 10 is at position 12367 and is: 45345034307070335134555666635024887773984040811638642621462405132584886144532377986073358145244201027037357938621772142209616405161627868105778015683798484008416515631723931456447477569347791478527857610769841230294963993979332743283338510233568324905959949579062456226347439170271217301636734446205747735713945863253433483740897523300848534809108461545232146005948850103219707387931993351213767427376889962909120228840233148459893174855648971003785113710072840958355449134583864883502737527932996232809536817702108606704972939587420565678604933039194329667673092895550055241161559979675050092624926868871458018343441582176596250529929191074291128267400687630170072609517611336780875630097770219247592957358603597479920082804823686786983637602905120758432634792571445674234736536086607803064981671382406890757026618258218813838974120950628944981905245622673666105003761865085749235468633696285737487359622886699738348673700526205671244526918181580594955298707753497194308839658138978679642619068841057090153286541341850948526490774435583437961002123092419531093235665933120984675385509883201479274589825686372618722516148263128067561905455373547546391052207792967353055080744047776986080724800254366264659407835575877760954125271799727336398809285045211625324747901193314444326911865397319722296088570348383225778250731285584101470972417783253384871308701259088227660489254359610314686851359252857300051920723671060874215404857586898535480326782181158608408279619195590515799061476749269801594625879703128842040583394154984407245336678544432608949614158673227077737666441262935879700412919345767447983556401804802974663896313057250911344831472616803645836814569048455850429162273597559955668906516180684124175673531692230282389747164102840354665242183576801399969588541389263384471603494896180265191824808290512910947744283695895248414399524148990184306294157590456646295768057794302761262994594722655055964096754993330351814041078876315865417737202330307734602330970139055634545867680529440237996308032770048686378187392782129596233133189016852266246661061281495413831799689680798229084372131339265032436345645554714807426037605589639494475684095551532755566945308404954024645746798191900979340826809913903596772706839217198094879854114314539133185622222803259348108969761096223170124626733975997197050299930954519458027451118459941271137375313026666466736293093998795637218208416937639900402972180842804696606039337369141035805923341690099055369556837577997854655210298104810520933189746485979124018196909319459534410139406110909222206154476323155751314166239864898017483972228615274429236864538159277543058839398436445287447883114413176678000824006252470021316902405765477155539855660847221656031691805464444896245205294585896596429456743028035184934461863654010366528689573439474363008455160522485449967670027033722278013317043478614293767140874291908415829941007066088704322634788409768471256565985280476650542596002212877354416976349052617941581234194408366031879100064003856581616095600774809166667678552494014963764538137429108976853046746079327814801533505923050861435172931240483219080490333386411655709499209676409734678148401497706097394309678808363917083429809690764137065474753611515665686465587299537880762699588786806388064113473982700024684504900602299235464951536685894424056502683548413627701664149406689523427968563342971845946890142203694025409999036915672932674775076177305732160054942888943862654196099307096880517864598688559999274834885952923776254724920526071660112996620623947730034924455037553195043344076284277868014382989931974908029499035370171767959005233662215283043876585561861314525586427883264270058319450472084164619032769031551576680710841970852322393037539325119666017996420868400106357872834174841151636016380407477204298535548979650626662451521208388882826961445471585692683081515085943594719122539526092112699608963277450836088209198655644116097290500948029821534837319698118213632136665010898032500699148625910693981318114217177275819944652811575951322137901737402106063154587879423128895000245926444726344972103433121968004238984107852115590751270399146488824187698443051260706281584394068239758982021239369875085477123234398148089271884961390297847264240658363997995669574653293259007333471580117358907760396222393711792653221029880957923342959879228299043166437440244751583666970116390226900111585349332814785514272218273380355024057390176759649953398455756889742471454303755506911727925831506750404598474949928168799378739227974329175403082516852308081709012535982487502050229124471983831128472775878806304759862969459429435276021291396028499943188538938799118003305779761837824757679798705125552062627109499973531037724930204716526740123383060533255126341151138114409083882474220727550189504138353942985034294716389262234667241206934164898746082763605571675460470098269794908360324625747888989968089590921522328937339714693262302234111258577347278529070921677616792503989869186375043102196295624700036419726681106495130783471156636071196116110061360911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Mathematica/Wolfram Language

<lang Mathematica>nums = HarmonicNumber[Range[15000]]; nums;; 20 LengthWhile[nums, LessEqualThan[#]] + 1 & /@ Range[10]</lang>

Output:
{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504}
{2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367}

Nim

Using floats

<lang Nim>import strformat

iterator h(): (int, float) = 

 ## Yield the index of the term and its value.
 var n = 1
 var r = 0.0
 while true:
   r += 1 / n
   yield (n, r)
   inc n

echo "First 20 terms of the harmonic series:" for (idx, val) in h(): 

 echo &"{idx:2}: {val}"
 if idx == 20: break

echo()

var target = 1.0 for (idx, val) in h(): 

 if val > target:
   echo &"Index of the first term greater than {target.int:2}: {idx}"
   if target == 10: break
   else: target += 1</lang>
Output:
 1: 1.0
 2: 1.5
 3: 1.833333333333333
 4: 2.083333333333333
 5: 2.283333333333333
 6: 2.45
 7: 2.592857142857143
 8: 2.717857142857143
 9: 2.828968253968254
10: 2.928968253968254
11: 3.019877344877345
12: 3.103210678210678
13: 3.180133755133755
14: 3.251562326562327
15: 3.318228993228994
16: 3.380728993228994
17: 3.439552522640758
18: 3.495108078196314
19: 3.547739657143682
20: 3.597739657143682

Index of the first term greater than  1: 2
Index of the first term greater than  2: 4
Index of the first term greater than  3: 11
Index of the first term greater than  4: 31
Index of the first term greater than  5: 83
Index of the first term greater than  6: 227
Index of the first term greater than  7: 616
Index of the first term greater than  8: 1674
Index of the first term greater than  9: 4550
Index of the first term greater than 10: 12367

Using big integers

Library: bignum

<lang Nim>import strformat import bignum

iterator h(): (int, Rat) = 

 var n = 1
 var r = newRat()
 while true:
   r += newRat(1, n)
   yield (n, r)
   inc n

echo "First 20 terms of the harmonic series:" for (idx, val) in h(): 

 echo &"{idx:2}: {val}"
 if idx == 20: break

echo()

var target = 1 for (idx, val) in h(): 

 if val > target:
   echo &"Index of the first term greater than {target:2}: {idx}"
   if target == 10: break
   else: inc target</lang>
Output:
 1: 1
 2: 3/2
 3: 11/6
 4: 25/12
 5: 137/60
 6: 49/20
 7: 363/140
 8: 761/280
 9: 7129/2520
10: 7381/2520
11: 83711/27720
12: 86021/27720
13: 1145993/360360
14: 1171733/360360
15: 1195757/360360
16: 2436559/720720
17: 42142223/12252240
18: 14274301/4084080
19: 275295799/77597520
20: 55835135/15519504

Index of the first term greater than  1: 2
Index of the first term greater than  2: 4
Index of the first term greater than  3: 11
Index of the first term greater than  4: 31
Index of the first term greater than  5: 83
Index of the first term greater than  6: 227
Index of the first term greater than  7: 616
Index of the first term greater than  8: 1674
Index of the first term greater than  9: 4550
Index of the first term greater than 10: 12367

PARI/GP

<lang parigp>h=0 for(n=1,20,h=h+1/n;print(n," ",h)) h=0; n=1 for(i=1,10,while(h<i,h=h+1/n;n=n+1);print(n-1))</lang>

Perl

<lang perl>use strict; use warnings; use feature 'say'; use Math::AnyNum ':overload'; use List::AllUtils 'firstidx';

my(@H,$n) = 0; do { ++$n and push @H, $H[-1] + 1/$n } until $H[-1] >= 10; shift @H;

say 'First twenty harmonic numbers as rationals:'; my $c = 0; printf("%20s", $_) and (not ++$c%5) and print "\n" for @H[0..19];

say "\nIndex of first value (zero based):"; for my $i (1..10) { 

   printf "  greater than %2d: %5s\n", $i, firstidx { $_ > $i } @H;

}</lang>

Output:
First twenty harmonic numbers as rationals:
                   1                 3/2                11/6               25/12              137/60
               49/20             363/140             761/280           7129/2520           7381/2520
         83711/27720         86021/27720      1145993/360360      1171733/360360      1195757/360360
      2436559/720720   42142223/12252240    14274301/4084080  275295799/77597520   55835135/15519504

Index of first value (zero based):
  greater than  1:     1
  greater than  2:     3
  greater than  3:    10
  greater than  4:    30
  greater than  5:    82
  greater than  6:   226
  greater than  7:   615
  greater than  8:  1673
  greater than  9:  4549
  greater than 10: 12366

Phix

Library: Phix/mpfr
requires("0.8.4")
include mpfr.e
integer n = 1, gn = 1, lim = iff(platform()=JS?8:10)
mpq hn = mpq_init_set_si(1)
sequence gt = {}
puts(1,"First twenty harmonic numbers as rationals:\n")
while gn<=lim do
    if n<=20 then
        printf(1,"%18s%s",{mpq_get_str(hn),iff(mod(n,5)?" ","\n")})
    end if
    if n=100 then
        printf(1,"\nOne Hundredth:\n%s\n\n",{mpq_get_str(hn)})
    end if
    if mpq_cmp_si(hn,gn)>0 then
        gt &= n
        gn += 1
    end if
    n += 1
    mpq_add_si(hn,hn,1,n)
end while
printf(1,"(one based) Index of first value:\n")
for i=1 to length(gt) do
    printf(1,"  greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])})
end for
Output:
First twenty harmonic numbers as rationals:
                 1                3/2               11/6              25/12             137/60
             49/20            363/140            761/280          7129/2520          7381/2520
       83711/27720        86021/27720     1145993/360360     1171733/360360     1195757/360360
    2436559/720720  42142223/12252240   14274301/4084080 275295799/77597520  55835135/15519504

One Hundredth:
14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

(one based) Index of first value:
  greater than  1:      2 (second term)
  greater than  2:      4 (fourth term)
  greater than  3:     11 (eleventh term)
  greater than  4:     31 (thirty-first term)
  greater than  5:     83 (eighty-third term)
  greater than  6:    227 (two hundred and twenty-seventh term)
  greater than  7:    616 (six hundred and sixteenth term)
  greater than  8:  1,674 (one thousand, six hundred and seventy-fourth term)
  greater than  9:  4,550 (four thousand, five hundred and fiftieth term)
  greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)

(The last two entries are too slow/beyond the reach of the first version of mpfr.js, but it's quick on the desktop)

using standard floats

integer n = 1, gn = 1
atom hn = 1
sequence gt = {}
puts(1,"First twenty harmonic numbers as fractions:\n")
while gn<=10 do
    if n<=20 then
        printf(1,"%18.15f%s",{hn,iff(mod(n,5)?" ","\n")})
    end if
    if n=100 then
        printf(1,"\nOne Hundredth: %18.15f\n\n",{hn})
    end if
    if hn>gn then
        gt &= n
        gn += 1
    end if
    n += 1
    hn += 1/n
end while
printf(1,"(one based) Index of first value:\n")
for i=1 to length(gt) do
    printf(1,"  greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])})
end for
{} = wait_key()
Output:
First twenty harmonic numbers as fractions:
 1.000000000000000  1.500000000000000  1.833333333333333  2.083333333333333  2.283333333333333
 2.450000000000000  2.592857142857143  2.717857142857143  2.828968253968254  2.928968253968254
 3.019877344877345  3.103210678210678  3.180133755133755  3.251562326562327  3.318228993228994
 3.380728993228994  3.439552522640758  3.495108078196314  3.547739657143682  3.597739657143682

One Hundredth:  5.187377517639621

(one based) Index of first value:
  greater than  1:      2 (second term)
  greater than  2:      4 (fourth term)
  greater than  3:     11 (eleventh term)
  greater than  4:     31 (thirty-first term)
  greater than  5:     83 (eighty-third term)
  greater than  6:    227 (two hundred and twenty-seventh term)
  greater than  7:    616 (six hundred and sixteenth term)
  greater than  8:  1,674 (one thousand, six hundred and seventy-fourth term)
  greater than  9:  4,550 (four thousand, five hundred and fiftieth term)
  greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)

Prolog

Works with: SWI Prolog

<lang prolog>main:- 

   print_harmonic_series(20),
   nl,
   nth_harmonic_number(100, T),
   Num is numerator(T),
   Denom is denominator(T),
   writef('100th harmonic number: %t/%t\n', [Num, Denom]),
   nl,
   print_first_harmonic_greater_than(10).

print_harmonic_series(N):- 

   writef('First %t harmonic numbers:\n', [N]),
   harmonic_first(H),
   print_harmonic_series(N, H).

print_harmonic_series(N, H):- 

   H = h(I, T),
   Num is numerator(T),
   Denom is denominator(T),
   writef('%3r. %t/%t\n', [I, Num, Denom]),
   (I == N, ! ; harmonic_next(H, H1), print_harmonic_series(N, H1)).

print_first_harmonic_greater_than(N):- 

   harmonic_first(H),
   print_first_harmonic_greater_than(1, N, H).

print_first_harmonic_greater_than(N, L, _):- 

   N > L,
   !.

print_first_harmonic_greater_than(N, L, H):- 

   H = h(P, T),
   (T > N ->
       writef('Position of first term >%3r: %t\n', [N, P]),
       N1 is N + 1
       ;
       N1 = N),
   harmonic_next(H, H1),
   print_first_harmonic_greater_than(N1, L, H1).

harmonic_first(h(1, 1)).

harmonic_next(h(N1, T1), h(N2, T2)):- 

   N2 is N1 + 1,
   T2 is T1 + 1 rdiv N2.

nth_harmonic_number(N, T):- 

   harmonic_first(H),
   nth_harmonic_number(N, T, H).

nth_harmonic_number(N, T, h(N, T)):-!. nth_harmonic_number(N, T, H1):- 

   harmonic_next(H1, H2),
   nth_harmonic_number(N, T, H2).</lang>
Output:
First 20 harmonic numbers:
  1. 1/1
  2. 3/2
  3. 11/6
  4. 25/12
  5. 137/60
  6. 49/20
  7. 363/140
  8. 761/280
  9. 7129/2520
 10. 7381/2520
 11. 83711/27720
 12. 86021/27720
 13. 1145993/360360
 14. 1171733/360360
 15. 1195757/360360
 16. 2436559/720720
 17. 42142223/12252240
 18. 14274301/4084080
 19. 275295799/77597520
 20. 55835135/15519504

100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

Position of first term >  1: 2
Position of first term >  2: 4
Position of first term >  3: 11
Position of first term >  4: 31
Position of first term >  5: 83
Position of first term >  6: 227
Position of first term >  7: 616
Position of first term >  8: 1674
Position of first term >  9: 4550
Position of first term > 10: 12367

Python

A generator function using fractions: <lang python>from fractions import Fraction

def harmonic_series(): 

   n, h = Fraction(1), Fraction(1)
   while True:
       yield h
       h += 1 / (n + 1)
       n += 1

if __name__ == '__main__': 

   from itertools import islice
   for n, d in (h.as_integer_ratio() for h in islice(harmonic_series(), 20)):
       print(n, '/', d)</lang>
Output:
1 / 1
3 / 2
11 / 6
25 / 12
137 / 60
49 / 20
363 / 140
761 / 280
7129 / 2520
7381 / 2520
83711 / 27720
86021 / 27720
1145993 / 360360
1171733 / 360360
1195757 / 360360
2436559 / 720720
42142223 / 12252240
14274301 / 4084080
275295799 / 77597520
55835135 / 15519504


Or alternatively, in terms of itertools.accumulate:

<lang python>Harmonic series

from fractions import Fraction from itertools import accumulate, count, islice from operator import add


  1. harmonic :: [Fraction]

def harmonic(): 

   Non finite stream of the terms
      of the Harmonic series.
   
   return accumulate(
       (1 / Fraction(x) for x in count(1)),
       add
   )


  1. ------------------------- TEST -------------------------
  2. main :: IO ()

def main(): 

   Tests of the harmonic series function

   print('First 20 terms of the harmonic series:')
   print('\n'.join([
       showFraction(nd) for nd in islice(harmonic(), 20)
   ]))

   print('\n100th term:')
   print(
       showFraction(
           next(islice(harmonic(), 99, None))
       )
   )

   print()
   print(
       'One-based indices of terms above threshold values:'
   )
   indexedHarmonic = enumerate(harmonic())
   print('\n'.join([
       next(
           showFirstLimit(n)(x) for x
           in indexedHarmonic if n < x[1]
       ) for n in range(1, 1 + 10)
   ]))


  1. ------------------ DISPLAY FORMATTING ------------------
  1. showFraction :: Fraction -> String

def showFraction(nd): 

   String representation of the fraction nd.
   
   n, d = nd.as_integer_ratio()

   return f'{n} / {d}'


  1. showFirstLimit :: Int -> (Int, Fraction) -> String

def showFirstLimit(n): 

   Report of 1-based index of first term
      with a value over n
   
   def go(indexedFraction):
       i = indexedFraction[0]

       return f'Term {1 + i} is the first above {n}'

   return go


  1. MAIN ---

if __name__ == '__main__': 

   main()</lang>
Output:
First 20 terms of the harmonic series:
1 / 1
3 / 2
11 / 6
25 / 12
137 / 60
49 / 20
363 / 140
761 / 280
7129 / 2520
7381 / 2520
83711 / 27720
86021 / 27720
1145993 / 360360
1171733 / 360360
1195757 / 360360
2436559 / 720720
42142223 / 12252240
14274301 / 4084080
275295799 / 77597520
55835135 / 15519504

100th term:
14466636279520351160221518043104131447711 / 2788815009188499086581352357412492142272

One-based indices of terms above threshold values:
Term 2 is the first above 1
Term 4 is the first above 2
Term 11 is the first above 3
Term 31 is the first above 4
Term 83 is the first above 5
Term 227 is the first above 6
Term 616 is the first above 7
Term 1674 is the first above 8
Term 4550 is the first above 9
Term 12367 is the first above 10

Quackery

<lang Quackery> [ $ "bigrat.qky" loadfile ] now! 

 0 n->v
 20 times 
   [ i^ 1+ n->v 1/v v+
     2dup 20 point$ echo$
     say " = " 
     2dup vulgar$ echo$ cr ] 
 2drop
 cr
 1 temp put
 0 n->v 1
 [ dup dip
     [ n->v 1/v v+
      temp share n->v 2over v< ]
   swap if 
     [ temp share echo
       say " : "
       dup echo cr 
       1 temp tally ]
   temp share 11 < while
   1+ 
   again ]
 temp release
 drop 2drop</lang>
Output:
1 = 1/1
1.5 = 3/2
1.83333333333333333333 = 11/6
2.08333333333333333333 = 25/12
2.28333333333333333333 = 137/60
2.45 = 49/20
2.59285714285714285714 = 363/140
2.71785714285714285714 = 761/280
2.82896825396825396825 = 7129/2520
2.92896825396825396825 = 7381/2520
3.01987734487734487734 = 83711/27720
3.10321067821067821068 = 86021/27720
3.18013375513375513376 = 1145993/360360
3.25156232656232656233 = 1171733/360360
3.31822899322899322899 = 1195757/360360
3.38072899322899322899 = 2436559/720720
3.43955252264075793488 = 42142223/12252240
3.49510807819631349043 = 14274301/4084080
3.54773965714368191148 = 275295799/77597520
3.59773965714368191148 = 55835135/15519504

1 : 2
2 : 4
3 : 11
4 : 31
5 : 83
6 : 227
7 : 616
8 : 1674
9 : 4550
10 : 12367

R

Direct Summation

The talk page helpfully points out that we can be remarkably lazy here. <lang rsplus>HofN <- function(n) sum(1/seq_len(n)) #Task 1 H <- sapply(1:100000, HofN) print(H[1:20]) #Task 2 print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</lang>

Output:
> print(H[1:20]) #Task 2
 [1] 1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968 3.019877 3.103211 3.180134 3.251562
[15] 3.318229 3.380729 3.439553 3.495108 3.547740 3.597740

> print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch
 [1]     2     4    11    31    83   227   616  1674  4550 12367

Cumulative Sums

As for doing this properly, R provides a handy cumsum function. <lang rsplus>firstNHarmonicNumbers <- function(n) cumsum(1/seq_len(n)) #Task 1 H <- firstNHarmonicNumbers(100000) #Runs stunningly quick print(H[1:20]) #Task 2 print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</lang>

Raku

Using Lingua::EN::Numbers from the Raku ecosystem. <lang perl6>use Lingua::EN::Numbers;

my @H = [\+] (1..*).map: { FatRat.new: 1, $_ };

say "First twenty harmonic numbers as rationals:\n", 

   @H[^20]».&pretty-rat.batch(5)».fmt("%18s").join: "\n";

put "\nOne Hundredth:\n", pretty-rat @H[99];

say "\n(zero based) Index of first value:"; printf " greater than %2d: %6s (%s term)\n", 

 $_, comma( my $i = @H.first(* > $_, :k) ), ordinal 1 + $i for 1..10;</lang>
Output:
First twenty harmonic numbers as rationals:
                 1                3/2               11/6              25/12             137/60
             49/20            363/140            761/280          7129/2520          7381/2520
       83711/27720        86021/27720     1145993/360360     1171733/360360     1195757/360360
    2436559/720720  42142223/12252240   14274301/4084080 275295799/77597520  55835135/15519504

One Hundredth:
14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

(zero based) Index of first value:
  greater than  1:      1 (second term)
  greater than  2:      3 (fourth term)
  greater than  3:     10 (eleventh term)
  greater than  4:     30 (thirty-first term)
  greater than  5:     82 (eighty-third term)
  greater than  6:    226 (two hundred twenty-seventh term)
  greater than  7:    615 (six hundred sixteenth term)
  greater than  8:  1,673 (one thousand, six hundred seventy-fourth term)
  greater than  9:  4,549 (four thousand, five hundred fiftieth term)
  greater than 10: 12,366 (twelve thousand, three hundred sixty-seventh term)

REXX

The default number of decimal digits (9) could've been used instead of   80   for this task's particular limits. <lang rexx>/*REXX pgm to calculate N numbers (sums) in the harmonic series and also when they > X. */ parse arg digs sums high ints /*obtain optional arguments from the CL*/ if digs= | digs="," then digs= 80 /*Not specified? Then use the default.*/ if sums= | sums="," then sums= 20 /* " " " " " " */ if high= | high="," then high= 10 /* " " " " " " */ if ints= | ints="," then ints= 1 2 3 4 5 6 7 8 9 10 /*Not specified? " " " */ w= length(sums) + 2 /*width of Nth harmonic index + suffix.*/ numeric digits digs /*have REXX use more numeric dec. digs.*/ 

                        s= 0                    /*initialize harmonic series sum to 0. */
     do j=1  for sums;  s= s + 1/j              /*calc  "sums" of harmonic series nums.*/
     @iter= right((j)th(j), w)                  /*obtain a nicely formatted sum index. */
     say right(@iter, w)  'harmonic sum ──►'  s /*indent the output to the terminal.   */
     end   /*j*/

say /*have a blank line between output sets*/ many= words(ints) /*obtain number of limits to be used. */ z= word(ints, 1) /* " the first " " " " */ lastInt= word(ints, many) /* " " last " " " " */ w= length(lastInt) /*W: is the maximum width of any limit*/

  1. = 1 /*a pointer to a list of integer limits*/
                   s= 0                         /*initialize harmonic series sum to 0. */
     do j=1;       s= s + 1/j                   /*calculate sums of harmonic sum index.*/
     if s<=z  then iterate                      /*Is sum <= a limit?  Then keep going. */
     iter= commas(j)th(j)                       /*obtain a nicely formatted sum index. */
     L= length(iter)                            /*obtain length so as to align output. */
     @iter= right(iter, max(L, 25) )            /*indent the output to the terminal.   */
     say @iter " iteration of the harmonic series, the sum is greater than "  right(z, w)
     #= # + 1                                   /*bump the pointer to the next limit.  */
     if #>many  then leave                      /*Are at the end of the limits?  Done. */
     z= word(ints, #)                           /*point to the next limit to be used.  */
     end   /*j*/                                /* [↑]  above indices are unity─based. */

exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? th: parse arg x; return word('th st nd rd', 1 + (x//10) *(x//100%10\==1) *(x//10<4))</lang>

output   when using the default inputs:
 1st harmonic sum ──► 1
 2nd harmonic sum ──► 1.5
 3rd harmonic sum ──► 1.8333333333333333333333333333333333333333333333333333333333333333333333333333333
 4th harmonic sum ──► 2.0833333333333333333333333333333333333333333333333333333333333333333333333333333
 5th harmonic sum ──► 2.2833333333333333333333333333333333333333333333333333333333333333333333333333333
 6th harmonic sum ──► 2.4500000000000000000000000000000000000000000000000000000000000000000000000000000
 7th harmonic sum ──► 2.5928571428571428571428571428571428571428571428571428571428571428571428571428571
 8th harmonic sum ──► 2.7178571428571428571428571428571428571428571428571428571428571428571428571428571
 9th harmonic sum ──► 2.8289682539682539682539682539682539682539682539682539682539682539682539682539682
10th harmonic sum ──► 2.9289682539682539682539682539682539682539682539682539682539682539682539682539682
11th harmonic sum ──► 3.0198773448773448773448773448773448773448773448773448773448773448773448773448773
12th harmonic sum ──► 3.1032106782106782106782106782106782106782106782106782106782106782106782106782106
13th harmonic sum ──► 3.1801337551337551337551337551337551337551337551337551337551337551337551337551337
14th harmonic sum ──► 3.2515623265623265623265623265623265623265623265623265623265623265623265623265623
15th harmonic sum ──► 3.3182289932289932289932289932289932289932289932289932289932289932289932289932290
16th harmonic sum ──► 3.3807289932289932289932289932289932289932289932289932289932289932289932289932290
17th harmonic sum ──► 3.4395525226407579348755819344054638172285231108760520525226407579348755819344055
18th harmonic sum ──► 3.4951080781963134904311374899610193727840786664316076080781963134904311374899611
19th harmonic sum ──► 3.5477396571436819114837690689083877938367102453789760291308278924377995585425927
20th harmonic sum ──► 3.5977396571436819114837690689083877938367102453789760291308278924377995585425927

                      2nd  iteration of the harmonic series, the sum is greater than   1
                      4th  iteration of the harmonic series, the sum is greater than   2
                     11th  iteration of the harmonic series, the sum is greater than   3
                     31st  iteration of the harmonic series, the sum is greater than   4
                     83rd  iteration of the harmonic series, the sum is greater than   5
                    227th  iteration of the harmonic series, the sum is greater than   6
                    616th  iteration of the harmonic series, the sum is greater than   7
                  1,674th  iteration of the harmonic series, the sum is greater than   8
                  4,550th  iteration of the harmonic series, the sum is greater than   9
                 12,367th  iteration of the harmonic series, the sum is greater than  10

Ring

<lang ring> decimals(12) sum = 0 nNew = 1 limit = 13000 Harmonic = []


for n = 1 to limit 

   sum += 1/n
   add(Harmonic,[n,sum])

next

see "The first twenty harmonic numbers are:" + nl for n = 1 to 20 

   see "" + Harmonic[n][1] + " -> " + Harmonic[n][2] + nl

next see nl

for m = 1 to 10 

   for n = nNew to len(Harmonic)
       if Harmonic[n][2] > m
          see "The first harmonic number greater than "
          see "" + m + " is " + Harmonic[n][2] + ", at position " + n + nl
          nNew = n
          exit
       ok
   next

next </lang>

Output:
The first twenty harmonic numbers are:
1 -> 1
2 -> 1.500000000000
3 -> 1.833333333333
4 -> 2.083333333333
5 -> 2.283333333333
6 -> 2.450000000000
7 -> 2.592857142857
8 -> 2.717857142857
9 -> 2.828968253968
10 -> 2.928968253968
11 -> 3.019877344877
12 -> 3.103210678211
13 -> 3.180133755134
14 -> 3.251562326562
15 -> 3.318228993229
16 -> 3.380728993229
17 -> 3.439552522641
18 -> 3.495108078196
19 -> 3.547739657144
20 -> 3.597739657144

The first harmonic number greater than 1 is 1.500000000000, at position 2
The first harmonic number greater than 2 is 2.083333333333, at position 4
The first harmonic number greater than 3 is 3.019877344877, at position 11
The first harmonic number greater than 4 is 4.027245195437, at position 31
The first harmonic number greater than 5 is 5.002068272680, at position 83
The first harmonic number greater than 6 is 6.004366708346, at position 227
The first harmonic number greater than 7 is 7.001274097134, at position 616
The first harmonic number greater than 8 is 8.000485571996, at position 1674
The first harmonic number greater than 9 is 9.000208062931, at position 4550
The first harmonic number greater than 10 is 10.000043008276, at position 12367

Wren

Library: Wren-big
Library: Wren-fmt

<lang ecmascript>import "/big" for BigRat import "/fmt" for Fmt

var harmonic = Fn.new { |n| (1..n).reduce(BigRat.zero) { |sum, i| sum + BigRat.one/i } }

BigRat.showAsInt = true System.print("The first 20 harmonic numbers and the 100th, expressed in rational form, are:") var numbers = (1..20).toList numbers.add(100) for (i in numbers) Fmt.print("$3d : $s", i, harmonic.call(i))

System.print("\nThe first harmonic number to exceed the following integers is:") var i = 1 var limit = 10 var n = 1 var h = 0 while (true) { 

   h = h + 1/n
   if (h > i) {
       Fmt.print("integer = $2d  -> n = $,6d  ->  harmonic number = $9.6f (to 6dp)", i, n, h)
       i = i + 1
       if (i > limit) return
   }
   n = n + 1

}</lang>

Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are:
  1 : 1
  2 : 3/2
  3 : 11/6
  4 : 25/12
  5 : 137/60
  6 : 49/20
  7 : 363/140
  8 : 761/280
  9 : 7129/2520
 10 : 7381/2520
 11 : 83711/27720
 12 : 86021/27720
 13 : 1145993/360360
 14 : 1171733/360360
 15 : 1195757/360360
 16 : 2436559/720720
 17 : 42142223/12252240
 18 : 14274301/4084080
 19 : 275295799/77597520
 20 : 55835135/15519504
100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272

The first harmonic number to exceed the following integers is:
integer =  1  -> n =      2  ->  harmonic number =  1.500000 (to 6dp)
integer =  2  -> n =      4  ->  harmonic number =  2.083333 (to 6dp)
integer =  3  -> n =     11  ->  harmonic number =  3.019877 (to 6dp)
integer =  4  -> n =     31  ->  harmonic number =  4.027245 (to 6dp)
integer =  5  -> n =     83  ->  harmonic number =  5.002068 (to 6dp)
integer =  6  -> n =    227  ->  harmonic number =  6.004367 (to 6dp)
integer =  7  -> n =    616  ->  harmonic number =  7.001274 (to 6dp)
integer =  8  -> n =  1,674  ->  harmonic number =  8.000486 (to 6dp)
integer =  9  -> n =  4,550  ->  harmonic number =  9.000208 (to 6dp)
integer = 10  -> n = 12,367  ->  harmonic number = 10.000043 (to 6dp)
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