Finite sum of the Leibniz formula for $\pi/4$ The Leibniz formula for $\pi/4$ is:
$$\sum_{i=0}^\infty \frac{(-1)^i}{2i+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... $$
How to calculate the summation of the first $n$ terms in the above series?

you can use the equivalent formula which is
$$1-\frac{1}{3}=\frac{2}{1*3}$$
$$\frac{1}{5}-\frac{1}{7}=\frac{2}{5*7}$$
and so on to get 
$$S=2(\frac{1}{1*3}+\frac{1}{5*7}+\frac{1}{9*11}+\frac{1}{13*15}+.....)$$ 
