Harmonic Number Calculator

Enter \( N \) to calculate the \( N \)th harmonic number.

Harmonic Number Calculate

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What Are Harmonic Numbers?

Harmonic numbers, often denoted as \( H_n \), represent the sum of the reciprocals of the first \( n \) natural numbers. The formula for the \( n \)th harmonic number is: \( H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) Harmonic numbers grow steadily as \( n \) increases, trending toward infinity, though the growth rate slows over time. They have significant applications in various fields, including physics, computer science, and mathematics.

How to Calculate the \( N \)th Harmonic Number

To calculate the \( N \)th harmonic number \( H_n \), simply sum the reciprocals of all integers from 1 to \( n \). The general formula is: \( H_n = \sum_{k=1}^{n} \frac{1}{k} \) This can be computed using recursive, iterative, or direct summation methods.

Examples

Example 1: Calculate the 5th Harmonic Number

Solution:

\( H_5 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = 2.2833333 \)

Example 2: Calculate the 10th Harmonic Number

Solution:

\( H_{10} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} = 2.929 \)

Example 3: Calculate the 20th Harmonic Number

Solution:

\( H_{20} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + \frac{1}{13} + \frac{1}{14} + \frac{1}{15} + \frac{1}{16} + \frac{1}{17} + \frac{1}{18} + \frac{1}{19} + \frac{1}{20} = 3.5977 \)

The First 100 Harmonic Numbers

H₁ = 1
H₂ = 1.5
H₃ = 1.83333
H₄ = 2.08333
H₅ = 2.28333
H₆ = 2.45
H₇ = 2.59286
H₈ = 2.71786
H₉ = 2.82897
H₁₀ = 2.92897
H₁₁ = 3.01988
H₁₂ = 3.10321
H₁₃ = 3.18013
H₁₄ = 3.25156
H₁₅ = 3.31823
H₁₆ = 3.38073
H₁₇ = 3.43955
H₁₈ = 3.49511
H₁₉ = 3.54774
H₂₀ = 3.59774
H₂₁ = 3.64536
H₂₂ = 3.69081
H₂₃ = 3.73429
H₂₄ = 3.77596
H₂₅ = 3.81596

H₂₆ = 3.85442
H₂₇ = 3.89146
H₂₈ = 3.92717
H₂₉ = 3.96165
H₃₀ = 3.99499
H₃₁ = 4.02725
H₃₂ = 4.0585
H₃₃ = 4.0888
H₃₄ = 4.11821
H₃₅ = 4.14678
H₃₆ = 4.17456
H₃₇ = 4.20159
H₃₈ = 4.2279
H₃₉ = 4.25354
H₄₀ = 4.27854
H₄₁ = 4.30293
H₄₂ = 4.32674
H₄₃ = 4.35
H₄₄ = 4.37273
H₄₅ = 4.39495
H₄₆ = 4.41669
H₄₇ = 4.43796
H₄₈ = 4.4588
H₄₉ = 4.47921
H₅₀ = 4.49921

H₅₁ = 4.51881
H₅₂ = 4.53804
H₅₃ = 4.55691
H₅₄ = 4.57543
H₅₅ = 4.59361
H₅₆ = 4.61147
H₅₇ = 4.62901
H₅₈ = 4.64625
H₅₉ = 4.6632
H₆₀ = 4.67987
H₆₁ = 4.69626
H₆₂ = 4.71239
H₆₃ = 4.72827
H₆₄ = 4.74389
H₆₅ = 4.75928
H₆₆ = 4.77443
H₆₇ = 4.78935
H₆₈ = 4.80406
H₆₉ = 4.81855
H₇₀ = 4.83284
H₇₁ = 4.84692
H₇₂ = 4.86081
H₇₃ = 4.87451
H₇₄ = 4.88802
H₇₅ = 4.90136

H₇₆ = 4.91451
H₇₇ = 4.9275
H₇₈ = 4.94032
H₇₉ = 4.95298
H₈₀ = 4.96548
H₈₁ = 4.97782
H₈₂ = 4.99002
H₈₃ = 5.00207
H₈₄ = 5.01397
H₈₅ = 5.02574
H₈₆ = 5.03737
H₈₇ = 5.04886
H₈₈ = 5.06022
H₈₉ = 5.07146
H₉₀ = 5.08257
H₉₁ = 5.09356
H₉₂ = 5.10443
H₉₃ = 5.11518
H₉₄ = 5.12582
H₉₅ = 5.13635
H₉₆ = 5.14676
H₉₇ = 5.15707
H₉₈ = 5.16728
H₉₉ = 5.17738
H₁₀₀ = 5.18738

Harmonic Number Calculator

Harmonic Number Calculate

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What Are Harmonic Numbers?

How to Calculate the \( N \)th Harmonic Number

Examples

Example 1: Calculate the 5th Harmonic Number

Example 2: Calculate the 10th Harmonic Number

Example 3: Calculate the 20th Harmonic Number

The First 100 Harmonic Numbers

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