Complex Summation Notation Question

Question

I would like to figure out how to solve summation notation that has variables on both numerator and denominator. The sample question is: $\sum_{n=2}^{100} \frac{n}{n^{2}-1}-\frac{1}{n}$

Answer 1

We'll use partial fractions, second differences & telescoping series. Let's define $a_n:=\frac{1}{n}-\frac{1}{n+1}$; we'll need it in a moment. Since $\frac{2n}{n^2-1}=\frac{1}{n-1}+\frac{1}{n+1}$, your sum is$$\begin{align}\frac12\sum_{n=2}^{100}\left(\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}\right)&=\frac12\sum_n(a_{n-1}-a_n)\\&=\frac{a_1-a_{100}}{2}\\&=\frac12\left(\frac12-\frac{1}{100}+\frac{1}{101}\right).\end{align}$$Feel free to rewrite that as you like.