# 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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### Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?

Question: Is there a closed-form expression for $$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$ where $\text{Ei}(z)$ is the exponential integral ...
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### Asymptotic behaviour of the $\sum_{k=3}^n \frac{n!}{(n-k)! n^k k}$ [duplicate]

We can show that $\sum_{k=3}^n \frac{n!}{(n-k)! n^k k} = \Theta(\ln(n))$ using the fact that $\sum_{k=3}^n \frac{1}{k} = \Theta(\ln(n))$ and $\frac{n!}{(n-k)! n^k} = \Theta(1)$ when $k=[\sqrt{n}]$. ...
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1 vote
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### Harmonic number denominator divisibilty proof

I'm looking for the complete proof (in a textbook preferably) of harmonic number $H_n$ denominator being divisible by all primes $p\in\left[\frac{n+1}{2},n\right]$. I found one by induction in ...
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### Show that there are no real solutions of $1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$ where $H_k = \sum_{i=1}^{k} \frac{1}{i}.$

[Edited] Show that there are no real solutions of $$1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$ I managed to prove this, and I want to ...
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1 vote
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### Upper limit for $(1+1/2)(1+1/3)\dots(1+1/n)$

I am trying to find an upper limit for the product $(1+1/2)(1+1/3)\dots(1+1/n)$. I tried using AM-GM inequality as follows: \begin{align} (1+1/2)(1+1/3)\dots(1+1/n) \leq \left(\frac{1}{n-1}\left( -1 + ...
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### Sums of harmonic series

If $$f(n, a, b)=\sum_{k_1=a+1}^{b}\sum_{k_2=a+1}^{k_1}\cdots\sum_{k_{2n}=a+1}^{k_{2n-1}}\frac{1}{k_1k_2\ldots k_{2n}}$$ $f(0, a, b)=1$ and $$g(a, b)=\sum_{n=0}^{\infty}f(n, a, b)$$ is there a way to ...
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### Approximating the Digamma function for small arguments

There are several ways to approximate the Digamma function $\psi(x)$ that become exact for $x\to\infty$. The simplest approximation is $$\lim_{x\to\infty}(\psi(x)-\mathrm{ln}(x))=0$$ There are other ...
I have been playing with the series which I had been calling the 'Double Basel problem' for the past couple of hours $$\sum_{n=1}^{\infty} \sum_{m=1}^\infty \frac{1}{{n^2 +m^2}}.$$ After wrestling ...