# 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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### Inequality $\sum\limits_{k=1}^n \frac{1}{n+k} \le \frac{3}{4}$

I recently came across the following exercise: Prove that $$\sum_{k=1}^n \frac{1}{n+k} \le \frac{3}{4}$$ for every natural number $n \ge 1$. I immediately tried by induction, by I did not ...
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### Proving that $~\sum\limits_{k=2}^{\infty}\frac{(-1)^k}{k^2}~H_k~H_{k-1}=\frac{3}{16}~\zeta(4)$

To show that $$\sum\limits_{k=2}^{\infty} \frac{(-1)^{k}}{k^{2}} \, \left(1+\frac{1}{2}+...+\frac{1}{k}\right) \cdot \left(1+\frac{1}{2}+...+\frac{1}{k-1}\right) = \frac{3}{16}\zeta(4).$$ I came ...
I had this question : Does the sequence \begin{align*} u_n=\sum_{i=1}^n\frac{1}{i} \end{align*} have a real end ? Well my teacher said no but the last question was Is \begin{align*} u_n=\... 1answer 65 views ### Prove that \sum_{n=1}^\infty\frac{\mu(n)}{n}H_n\sum_{k=n+1}^\infty\frac{\mu(k)}{k^2} is convergent For integers n\geq 1 letH_n=1+\frac{1}{2}+\ldots+\frac{1}{n}$$the nth harmonic number, and \mu(n) the Möbius function. See, if you need it, this Wikipedia to know the definition of Möbius ... 5answers 467 views ### Closed form for {\large\int}_0^1\frac{\ln^4(1+x)\ln x}x \, dx Can someone compute$$ \int_0^1\frac{\ln^4(1+x)\ln x}x \,dx$$in closed form? I conjecture that the answer can be expressed as a polynomial function with rational coefficients in constants of the ... 2answers 232 views ### Elementary way to calculate the series \sum\limits_{n=1}^{\infty}\frac{H_n}{n2^n} I want to calculate the series of the Basel problem \displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^2}} by applying the Euler series transformation. With some effort I got that$$\displaystyle{\frac{\...
Let us define a following generating function: $${\bf H}^{(1,1,1)}_n(x) := \sum\limits_{m=1}^\infty [H_m]^3 \cdot \frac{x^m}{m^n}$$ Now, by using results from Generating ...