# 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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### Can $\int_0^\infty f (x) \, dx$ exist if $\lim_{x \to \infty} f(x)$ does not exist?

Is is possible to have a function for which $\lim_{x \to \infty} f(x)$ does not exist, but $\int_0^\infty f(x) \, dx$ exists and is finite? I think I've found an example actually, but I'm not sure it ...
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### $\zeta(4)$ in terms of a series of $\zeta(3)$ and harmonic numbers

The other day I believe I found a proof that: $$\sum_{k=1}^\infty \frac{\zeta(2)-H_k^{(2)}}{k} = \zeta(3)$$ I was wondering if a general recursion like this was well known, but I couldn't find ...
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### How was the harmonic number's relation to the hurwitz zeta function derived?

How was the generalized harmonics number's relation to the hurwitz zeta function derived? $H_{n,\ m} =\zeta ( m,\ 1) -\zeta ( m,\ n+1),\ \Re(m)>1$ I tried looking at the series representations for ...
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### How to Evaluate the Sum $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\left(H_{n}\right)^2}{2n+1}$

How can I evaluate the Sum : $$S=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\left(H_{n}\right)^2}{2n+1}$$ where $\left(H_{n}\right)^2$ denotes a Harmonic Number Squared. The sum converges and is approximated ...
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### Upper bound of $\sum_{k=1}^n \frac{1}{\sqrt{k}}$?

I am looking for an upper bound of $\sum_{k=1}^n \frac{1}{\sqrt{k}}$. Alternatively, is the sequence $\frac{1}{n\sqrt{n}}\sum_{k=1}^n \frac{1}{\sqrt{k}}$ bounded? I am trying to use a Strong law of ...
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103 views

### Questions regarding $\ln(x) = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}(\zeta(n,x)-\zeta(n))$. Have I found something “new”?
Introduction TL;DR I was messing around with the Taylor series for $\ln(x)$ when I ended up with the formula \begin{align} \ln(x) &= \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}(\zeta(n,x)-\zeta(n)) \\\...