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 second harmonic number.


Proof

\(\displaystyle H_{10000}\) \(\approx\) \(\displaystyle \ln 10 \, 000 + \gamma + \dfrac 1 {2 \times 10000} - \dfrac 1 {12 \times \left({10000}\right)^2} + \dfrac 1 {12 \times \left({10000}\right)^4} + \epsilon\) Approximate Size of Sum of Harmonic Series
\(\displaystyle \) \(=\) \(\displaystyle 4 \ln 10 + \gamma + \dfrac 1 {2 \times 10000} - \dfrac 1 {12 \times \left({10000}\right)^2} + \dfrac 1 {12 \times \left({10000}\right)^4} + \epsilon\) Logarithm of Power

$\blacksquare$

We have:

\(\displaystyle \ln 10\) \(\approx\) \(\displaystyle 2 \cdotp 30258 \, 50929 \, 94045 \, 68401 \, 7991 \ldots\) Natural Logarithm of 10
\(\displaystyle \gamma\) \(\approx\) \(\displaystyle 0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \ldots\) Definition of Euler-Mascheroni Constant
\(\displaystyle \dfrac 1 {2 \times 10000}\) \(=\) \(\displaystyle 0 \cdotp 00005\) Definition of Euler-Mascheroni Constant
\(\displaystyle \dfrac 1 {12 \times 10000^2}\) \(\approx\) \(\displaystyle 0 \cdotp 00000 \, 00008 \, 33333 \, 33333\)
\(\displaystyle \dfrac 1 {120 \times 10000^4}\) \(<\) \(\displaystyle 10^{-18}\)

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