Tag Archives: polylogarithm

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 →

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 →