Calculate the sum of the following infinite sequence: $\sum\limits_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)}$

Question

How to calculate the sum of the following infinite sequence? $$\sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)} = ?$$

Answer 1

Hint:

There exist real numbers $A$ and $B$ such that

$$\frac{1}{(2n-1)(2n+1)} = \frac{A}{2n-1} + \frac{B}{2n+1}$$

$\frac{1}{(2n-1)(2n+1)} = \frac{0.5}{2n-1} - \frac{0.5}{2n+1}$

See if anything cancels nicely.

If you are still having trouble, search for the phrase "telescoping series" for more information on how to continue.