Monthly Archives: July 2020
Euler Sums involving square of Harmonic numbers
In my previous post on Euler sums, we evaluated sums containing $H_n$ and $H_n^{(2)}$. In this post, we’ll derive some further results using the integral $\int_0^x \frac{\log^3(1-t)}{t}dt$. Our starting point is the following generating function identity: $$ \sum_{n=1}^\infty (H_n)^2 x^n … Continue reading
The Contour Integration approach to Infinite Series
In this post, we will evaluate the series $\sum_{n=0}^\infty \frac{\cot\left(\frac{2n+1}{2}\pi\sqrt{2} \right)}{(2n+1)^3}$ using contour integration. Let’s define $f:\mathbb{C}\to \mathbb{C}$ as $$ f(z) = \frac{\pi \tan(\pi z)\tan(\pi z \theta)}{z^3} $$ where the parameter $\theta$ is a positive irrational number. Let $C_N$ denote … Continue reading
An alternating Euler Sum
In this post, we will evaluate the famous Euler sum: $$\sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^3}\quad \color{blue}{\cdots (*)} $$ where $H_n = \sum_{k=1}^n \frac{1}{k}=\int_0^1 \frac{1-t^n}{1-t}dt$ is the $n$-th harmonic number. This series resists contour integration techniques which makes it’s computation quite challenging. We will … Continue reading