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?

$$\sum_{i=0}^n\frac{(-1)^i}{2i+1}=\frac{(-1)^n}{2}[\psi_0(3/2+n)-\psi_0(1/2)],$$
where $\psi_0(x)$ is the Polygamma function (also known as the DiGamma function for this case). There are likely no further simplifications possible (if this even counts as one). 
