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.

582 questions
155 views

How to prove this question by Ramanujan?

click here for photo $$1+2\sum_{k=1}^\infty \frac{1}{(4k)^3-(4k)}= \frac{3}{2}\ln(2)\,.$$ well i have attatched a photo which has been asked to prove without using calculus,but how to solve this ...
93 views

59 views

64 views

Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, let $\mu(n)$ the Möbius function, see this MathWorld, and let $H_n$ the $n$th harmonic number as $H_n=1+1/2+\ldots+1/n$. Also we denote the Riemann zeta function with $\zeta(s)$...
51 views

Limit of the difference between two harmonic numbers

I am trying to evaluate the following limit: $$\lim_{n \rightarrow \infty} H_{kn} - H_n = \lim_{n \rightarrow \infty} \sum_{i = 1}^{(k - 1)n} \frac{1}{n + 1}$$ I know that the answer is $\ln k$, ...
38 views

Simple formula for $H_n = m + \alpha$?

Let $H_i$ be the $i$ th harmonic number. For a given positive integer $m$ we want to find the smallest possible positive integer value $n$ such that $H_n = m + \alpha$, where $\alpha > 0$. Let us ...
44 views

On integer sequences of the form $\sum_{n=1}^N (a(n))^2H_n^2,$ where $H_n$ is the $n$th harmonic number: refute my conjecture and add yourself example

After I've read a problem posed by Édouard Lucas from the first paragraph of the biography from this Wikipedia's article I am trying to get different variations of this problem using number theoretic ...
37 views

A non obvious example of a sequence $a(k)\cdot H_{b(k)}$ whose general term is integer many times, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, $H_n$ denotes the $n$th harmonic number. I am looking examples of arithmetic function $a(k)$ and $b(k)$, whose terms are integers $\geq 1$ and such that the terms of this ...
112 views

296 views

On twisted Euler sums

An interesting investigation started here and it showed that $$\sum_{k\geq 1}\left(\zeta(m)-H_{k}^{(m)}\right)^2$$ has a closed form in terms of values of the Riemann $\zeta$ function for any ...
78 views

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 $H_{n+\frac{1}{2}}$?