Is there a partial sum formula for the Harmonic Series?

Question

There is a partial sum formula for $$\sum_{x=1}^n x^1 = \frac{n(n+1)}{2}$$ and even one when the exponent of $x$ is $0$: $$\sum_{x=1}^n x^0 = n$$ but I cannot find one for exponent $-1$: $$\sum_{x=1}^n x^{-1} = ?$$

I tried $$\frac2{n(n+1)},$$ but that failed miserably.

Able to help on a related question: https://math.stackexchange.com/q/4546174/322359 ? — user3108815, Oct 07 '22 at 21:07

score 32 · Accepted Answer · answered Sep 17 '13 at 06:08

32

No, there is no nice closed form for the harmonic numbers. There are some very accurate approximations that are easily computed;

$$H_n\approx\ln n+\gamma+\frac1{2n}-\frac1{12n^2}$$

is quite good, where $\gamma\approx 0.5772156649$ is the Euler-Mascheroni constant.

answered Sep 17 '13 at 06:08

Brian M. Scott

616,228

Could you provide a source for that approximation please? – zerosofthezeta Sep 17 '13 at 06:35
@zerosofthezeta: It’s at the link, at the bottom of the Calculation section of the article. – Brian M. Scott Sep 17 '13 at 06:36
Got it, thanks! – zerosofthezeta Sep 17 '13 at 06:37
@zerosofthezeta: You’re welcome! – Brian M. Scott Sep 17 '13 at 06:38
1

I've found a better approximation (for my purposes) using: ln(x + 0.5) + 0.5772156649 – trr Jun 01 '16 at 02:18
This link in wiki is giving a closed form for $H_n$! Am I right? But it looks much worse than the Harmonic series itself! :) – Hosein Rahnama Jul 13 '16 at 00:11
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@H.R.: I wouldn't call either the integral representation or the alternating summation a closed form, but they do give alternative ways to look at $H_n$ that can be useful. – Brian M. Scott Jul 13 '16 at 01:53
Interesting. I have to look into this. Why wouldnt a finite series have a closed form? – CogitoErgoCogitoSum Jul 15 '17 at 17:34
@Cogito, because a lot of times they really don't have one and thus you have to invent them. Consider e.g. $\sum_{k=1}^n k^k$. – J. M. ain't a mathematician Jul 26 '17 at 05:00
Saying they dont have one is contradictory to saying you have to invent one. If one can be invented then it does in fact have one. – CogitoErgoCogitoSum Jul 26 '17 at 05:28
Anyone able to help answered a related question? https://math.stackexchange.com/q/4546174/322359 – user3108815 Oct 07 '22 at 21:06

Is there a partial sum formula for the Harmonic Series?

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