% m-file contains an example of useful functions using

% Compute 5!

n = 10;
prod(1:n);

% compute sum of ten first numbers

sum(1:n)

% compute the vector containing subsequent products of its elements

cumprod(1:n)

% compute the vector containing subsequrent sums of its elements

cumsum(1:n)

% compute value of sinus(k) (from its definition!)
% x - x^3/3! + x^5/5! + ... + (-1)^n x^(2n + 1)/ (2n + 1)!

% a bit of math: what kind of power series is it?

% odd powers so we need construct a vactor of odd values
% n from the definition
n = 10;
n1 = 2*[0:n] + 1;

% compute vector of (2n + 1)!
t = cumprod(1:(2*n + 1));

% compute vector of 1/(2n+1)!
r = 1./t(1:2:end);

% point of computation
k = 2;

% k to different powers defined by p
p = k.^n1

% how about signs?
s = (-1).^[2:(n+2)]

% and together

output = cumsum(s.*p.*r)

% exact value
output2 = sin(k)

% to plot
% How do you describe the convergence of the "output" series?

plot([0:n], output, 'ro', [0:n], output2*ones(1, n+1), 'b-');
title(['Sinus for k = ', num2str(k)]);
xlabel('n');
ylabel(['sin(', num2str(k), ')']);

% as it is seen from the example above the prod, sum, cumsum and cumprod functions can be successfully applied to power expressions