# 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.

591 questions
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### Harmonic Number for $H_{n+1}$ and $H_{n+\frac{1}{2}}$

According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$. How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$?
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### Limit Involving Generalised Harmonic Numbers

Is there an easy/straight-forward way to evaluate the limit $$\lim_{n\to\infty} \frac{\pi^4-90H_n^{(4)}}{4\pi^2(\pi^2-6H_n^{(2)})},$$ where $H_{n}^{(r)}$ denotes the $n$th generalised Harmonic number ...
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### About the partition of some reciprocals

Does there exists a partition of the reciprocals $\frac{1}{2}, \frac{1}{3},\cdots,\frac{1}{12}$, into two sets $A$ and $B$ such that $\displaystyle\sum_A\frac{1}{n} -\sum_B\frac{1}{n}=1$?
### Let $x_n$ denote the smallest integer such that $\sum\limits_{k=n}^{x_n} \frac 1k >1$, then $x_n\sim en$
For $n\geq 2$, let $x_n$ denote the smallest integer such that $\displaystyle \sum_{k=n}^{x_n} \frac 1k >1$. Prove that $\lim_n \frac{x_n}{n}\sim e$. I've found this problem in an old handout of ...