Harmonic Numbers/Examples/H10000

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
We have:

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.