# 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.

155 questions
22answers
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### Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ doesn't converge, on the other hand it grows very slowly?...
12answers
5k views

### The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

What is the sum of the 'second half' of the harmonic series? $$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial ...
10answers
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### Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
5answers
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1answer
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### Sum of Squares of Harmonic Numbers

Let $H_n$ be the $n^{th}$ harmonic number, $$H_n = \sum_{i=1}^{n} \frac{1}{i}$$ Question: Calculate the following $$\sum_{j=1}^{n} H_j^2.$$ I have attempted a generating function approach but ...
1answer
574 views

4answers
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2answers
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### Closed form expressions for harmonic sums

It is well known that there are deep connections between harmonic sums (discrete infinite sums that involve generalized harmonic numbers) and poly-logarithms. Bearing this in mind we have calculated ...