Apologies if this is answered somewhere else.
I am trying to understand how the author manipulated this series to get the general form. Can anyone show me the steps and provide the rules used?
$$S_n = \sum_{i=2}^n\frac{1}{i^2 - 1} = \frac{3}{4} - \frac{1}{2n} - \frac{1}{2(n + 1)}$$
Thank you!
$$S_n = \sum_{i=2}^n\frac{1}{i^2 - 1} = \sum_{i=2}^n \dfrac12\left(\dfrac1{i-1}-\dfrac1{i+1}\right).$$
This is a telescoping sum:
$$ \dfrac12\left[\left(\dfrac1{1}-\dfrac13\right)+\left(\dfrac1{2}-\dfrac14\right)+\left(\dfrac1{3}-\dfrac15\right)+\cdots+ \left(\dfrac1{n-2}-\dfrac1{n}\right)+\left(\dfrac1{n-1}-\dfrac1{n+1}\right)\right]$$
$$=\dfrac12\left[\dfrac1{1}+\dfrac12-\dfrac1{n}-\dfrac1{n+1}\right].$$
