Thanks for the help, I did what you said and I got the sum of both series:
by proving its a geometric series ...
What do you mean by separating real and imaginary parts, How can I do it here? and will it give me the sum of the original series:
sinx+sin2x+...+sinnx , even if i changed it to a kind of a complex seris?
It will give you the sum of BOTH series, cosines and sines:
And there you have the right side divided in real and imaginary part. Now use a little trigonometry to deduce that the real part is ,
and the imaginary part is .
OTOH, in the left side we have the sum , so again separate in real and imaginary parts this sum.