# find formulas

• November 7th 2009, 05:17 AM
find formulas
Find closed formulas for the following two sums:
1 + cos
X + cos 2X + . . . + cos nX

sin
X + sin 2X + . . . + sinnX

Hint: Use the sum of the geometric series with the general term eikX.
• November 7th 2009, 08:00 AM
tonio
Quote:

Find closed formulas for the following two sums:

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

sin
X + sin 2X + . . . + sinnX
Hint: Use the sum of the geometric series with the general term eikX.

The given hint is a huge one:

$\sum\limits_{k=0}^ne^{kxi}=\frac{e^{(n+1)xi}-1}{e^{xi}-1}$

Now just use that $e^{ir}=\cos r+i\sin r\;\;\forall\,r\in \mathbb{R}$ , plug in and separate real and imaginary parts.

Tonio
• November 7th 2009, 11:02 AM
Quote:

Originally Posted by tonio
The given hint is a huge one:

$\sum\limits_{k=0}^ne^{kxi}=\frac{e^{(n+1)xi}-1}{e^{xi}-1}$

Now just use that $e^{ir}=\cos r+i\sin r\;\;\forall\,r\in \mathbb{R}$ , plug in and separate real and imaginary parts.

Tonio

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 ...

$\sum= \frac{cos[(n+1)x] + isin[(n+1)x] -1}{cosx + isinx -1}$

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,
• November 7th 2009, 01:55 PM
tonio
Quote:

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 ...

$\sum= \frac{cos[(n+1)x] + isin[(n+1)x] -1}{cosx + isinx -1}$

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,

It will give you the sum of BOTH series, cosines and sines:

$\frac{\cos (n+1)x+i\sin (n+1)x-1}{\cos x+i\sin x-1}$ $=\frac{(\cos (n+1)x-1)+i\sin (n+1)x}{(\cos x-1)+i\sin x}\,\frac{(\cos x-1)-i\sin x}{(\cos x-1)-i\sin x}=$

$=\frac{\cos (n+1)x(\cos x-1)+\sin (n+1)x\sin x}{(\cos x-1)^2+\sin^2x}$ $-\frac{\cos (n+1)x\sin x-\sin (n+1)x(\cos x-1)}{(\cos x-1)^2+\sin^2x}\,i$

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 $\frac{\cos nx-\cos (n+1)x}{2(1-\cos x)}$ ,

and the imaginary part is $\frac{\sin nx - \sin (n+1)x}{2(1-\cos x)}$.

OTOH, in the left side we have the sum $\sum\limits_{k=1}^ne^{kxi}=\sum\limits_{k=0}^n(\co s kx+i\sin kx)$ , so again separate in real and imaginary parts this sum.

Tonio