Simplifying a fractional summation

Question

I'm trying to simplify the following:

$\sum_{i = 1}^{n} \frac{1}{(2i-1)(2i+1)}$

into

$\frac{n}{2n+1}$

How should I proceed?

Answer 1

Notice that

$$\frac1{(2i-1)(2i+1)}=\frac12\left(\frac1{2i-1}-\frac1{2i+1}\right)$$

This gives us the following telescoping series:

$$\sum_{i=1}^n\frac1{(2i-1)(2i+1)}=\sum_{i=1}^n\frac12\left(\frac1{2i-1}-\frac1{2i+1}\right)=\frac12\left(1-\frac1{2n+1}\right)=\frac n{2n+1}$$