Harmonic Numbers/Examples/H10000
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Example of Harmonic Number
To $15$ decimal places:
- $H_{10000} \approx 9 \cdotp 78760 \, 60360 \, 44382 \, \ldots$
where $H_{10000}$ denotes the $10 \, 000$th harmonic number.
Proof
| \(\ds H_{10000}\) | \(\approx\) | \(\ds \ln 10 \, 000 + \gamma + \dfrac 1 {2 \times 10000} - \dfrac 1 {12 \times \left({10000}\right)^2} + \dfrac 1 {12 \times \left({10000}\right)^4} + \epsilon\) | Approximate Size of Sum of Harmonic Series | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \ln 10 + \gamma + \dfrac 1 {2 \times 10000} - \dfrac 1 {12 \times \left({10000}\right)^2} + \dfrac 1 {12 \times \left({10000}\right)^4} + \epsilon\) | Logarithm of Power |
$\blacksquare$
We have:
| \(\ds \ln 10\) | \(\approx\) | \(\ds 2 \cdotp 30258 \, 50929 \, 94045 \, 68401 \, 7991 \ldots\) | Natural Logarithm of 10 | |||||||||||
| \(\ds \gamma\) | \(\approx\) | \(\ds 0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \ldots\) | Definition of Euler-Mascheroni Constant | |||||||||||
| \(\ds \dfrac 1 {2 \times 10000}\) | \(=\) | \(\ds 0 \cdotp 00005\) | Definition of Euler-Mascheroni Constant | |||||||||||
| \(\ds \dfrac 1 {12 \times 10000^2}\) | \(\approx\) | \(\ds 0 \cdotp 00000 \, 00008 \, 33333 \, 33333\) | ||||||||||||
| \(\ds \dfrac 1 {120 \times 10000^4}\) | \(<\) | \(\ds 10^{-18}\) |
Thus for an accuracy of $15$ decimal places it is unnecessary to consider $\dfrac 1 {120 \times 10000^4}$ and smaller terms.
Then:
2.30258 50929 94045 68401 x 4 ------------------------- 9.21034 03719 76182 73604 + 0.57721 56649 01532 86060 + 0.00005 ------------------------- 9.78760 50368 77715 59664 - 0.00000 00008 33333 33333 ------------------------- 9.78760 50360 44382 26331
Hence the result.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $5$