Algebra
# Partial Fractions

$\frac {1}{1 \times 2} + \frac {1}{1 \times 5} + \frac {1}{3 \times 3} + \frac {1}{2 \times 7} + \frac {1}{5 \times 4} + \ldots$

Find the sum of this infinite series.

$\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{2015^{2^{n}} - 2015^{-(2^{n})}}$

If the closed form of the series above is in the form $\frac a b$, where $a$ and $b$ are positive coprime integers, then find $b - a.$

For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$ If $\sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, find $a+b$.

This problem is shared by Sandeep S.