# Questions tagged [harmonic-numbers]

For questions regarding harmonic numbers, which are partial sums of the harmonic series. The $N$-th harmonic number is the sum of reciprocals of the first $N$ natural numbers.

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### Closed form of the sum $s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}$

I am interested to know if the following sum has a closed form $$s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}\tag{1}$$ I stumbled on this question while studying a very useful book about ...
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### Where is my mistake $\int_{0}^{1}\frac{\log(1-x)\log(1-x^2)}{x}dx=\frac{7}{4}\zeta(3)$

Edit As commented bellow by @Donald Splutterwit and @ Elliot Yu, it seems that my computation is numerically correct and the post is wrong! I also added a corollary from this computation. I saw the ...
0answers
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### Evaluating the integral: $\int _0^1\frac{\ln \left(x\right)\ln ^2\left(\frac{1-x}{1+x}\right)\operatorname{Li}_2\left(x\right)}{x}\:dx$

I'd like to evaluate and find the closed form for $$\int _0^1\frac{\ln \left(x\right)\ln ^2\left(\frac{1-x}{1+x}\right)\operatorname{Li}_2\left(x\right)}{x}\:dx$$ But it's very difficult since its ...
1answer
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### Proving $\sum_{n=1}^{\infty}\dfrac{H^{(a)}_{n}}{n^b} = \zeta(a,b) + \zeta(a+b)$

I've come across the formula: $$\sum_{n=1}^{\infty}\dfrac{H^{(a)}_{n}}{n^b} = \zeta(a,b) + \zeta(a+b)$$ where $H^{(a)}_{n}$ is the Generalized Harmonic nuber and $\zeta(a,b)$ is the Multiple zeta ...
1answer
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### Connection between $J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1}$ and $f(x)=\sum_{n=1}^\infty e^{-n^x} ?$

Consider the geometric series$$J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1}$$ I'm wondering if this has any connection with: $$f(x)=\sum_{n=1}^\infty e^{-n^x}.$$ $J_k$ can be interpreted as adding ...
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### Calculate $\sum_{k=1}^{\infty}\frac{|\sin(k x)|}{1+k^2}$

In https://math.stackexchange.com/a/4015346/198592 it was shown that the sum $$s(x) = \sum_{k=1}^{\infty}\frac{\sin(k x)}{1+k^2}$$ is exactly expressible in terms of the hypergeometric function. I ...
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### Product of Generalized Harmonic Numbers

I'm trying to sum two indipendent random variabile, in particular two zipf's law, mathematica say that the product of two generalized harmonic number $H_{p,s}$ and $H_{t,k}$, $H_{p,s}H_{t,k}$ is eqaul ...
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1answer
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### Can we identify the pole of the Gamma function with the limit of the harmonic numbers?

The expansion of the gamma function around $x=0$ is $$\Gamma(x)=\frac{1}{x}-\gamma+O(x).$$ The Euler constant $\gamma$ is defined by $$\gamma=\lim_{n\to\infty}(H_n-\log n)$$ where $H_n$ is the $n$th ...
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### Another bizarre sum involving a binomial coefficient and inverse powers of integers.

In the attempt to answer Binomial identity involving Harmonic numbers we stumbled on the following problem. Let $i\ge 0$ and $k \ge i+2$ and $p \ge 1$ be integers. Consider a following sum: \begin{...
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### Divergence of Harmonic Series

I understand that the divergence of the harmonic series is a classic proof but what I don't understand is the way it seems to contradict standard methods of finding if a series converges or not. For ...
1answer
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