How to calculate $\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$ How can I calculate the sum of the following infinite series: $$\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$$
Hint : $$\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)} = \sum_{n=1}^\infty \left (\frac{1}{2n-1}-\frac{1}{2n+1} \right)$$
This is how you do it formally:

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

Now take the limit $$\lim_{N\to\infty} \left(1 - \frac{1}{2N+1}\right) =1$$ 
So the series converges towards 1.