% Computes pi using the Taylor series for the arctan(x) at x=1:
% pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... + (-1)^N/(2N+1)

% Set the value of N
N = 1000;

% Initialize the variable used for the partial sums.
p = 0;

% Here is the implementation using a loop to sum the individual
% terms.
for k = 0:N
   p = p + (-1)^k/(2*k + 1);
end
p = 4*p;  % p contains the approximation to pi

% The above code can be "vectorized" as follows:
% (note: just uncomment out the following two lines.)
% k = 0:N;
% p = 4*sum((-1).^k./(2*k + 1));