arctan(1)=pi/4 wich means : the arc wich has 1 as tan is equal to pi/4 radians.
SEries expansion gives :
arctan(1) = 1-1/3+1/5-1/7....
so pi/4 = 1-1/3+1/5-1/7....
1-1/3+1/5-1/7....can be easily computed.
The series is the Geogory-Leibniz series. It is one of the slowest convergent series for pi. You need about to sum 300 terms to get 3.14 accurately, to get 10 digits you need about 10 billion terms and to get to 1000 digits you are in the realms of needing more computations than atoms in the universe (by a long way!).
I note that there are some transforms that get the convergence quickly (see wolfram site) , if that was the intention of the question, fine, but otherwise it is a very stupid question.