Sum $\sum_{i=1}^\infty \frac{2}{(2i +1)(2i+3)}$

Question

how can I calculate this: $$\sum_{i=1}^\infty \frac{2}{(2i +1)(2i+3)}$$ I don't have seen analysis yet, this is a olympic problem of my country.

Answer 1

Hint

Find $A,B\in\mathbb R$ s.t. $$\frac{1}{(2i+1)(2i+3)}=\frac{A}{2i+1}+\frac{B}{2i+3}.$$

Answer 2

$$=\sum_{i=1}^{\infty} (\frac {1}{2i+1} - \frac{1}{2i+3})$$

So It is $$\frac {1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}......$$

answer is $\frac {1}{3}$