Find closed formulas for the following two sums:

1 + cosX + cos 2X + . . . + cos nX

sinX + sin 2X + . . . + sinnX

Hint: Use the sum of the geometric series with the general term eikX.

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- November 7th 2009, 06:17 AMohadchfind formulasFind closed formulas for the following two sums:

1 + cosX + cos 2X + . . . + cos nX

sinX + sin 2X + . . . + sinnX

Hint: Use the sum of the geometric series with the general term eikX.

- November 7th 2009, 09:00 AMtonio
- November 7th 2009, 12:02 PMohadch
Hi Tonio,

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?

Thanks,

Ohad. - November 7th 2009, 02:55 PMtonio

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.

Tonio