Summation and Patterns Question

Question

I have a really urgent question. Today i was looking at this Pattern (Using fractions) : $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+\frac{1}{7*9}+...$=? The pattern is $\frac{n}{2n+1}$. I was wondering if you could do a summation formula, in which the end term is infinite. Thanks for the help!

Answer 1

Hint: This is a telescoping series. The general term is $\frac{1}{(2n+1)(2n-1)}$

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

I think you can continue after that