Sum of Series in |z| < 1: Sin(\frac{2\pi}{3})+z

[FanofAFan]

Homework Statement

Find the sum in |z| < 1 of the series sin($$\frac{2\pi}{3}$$) + z sin($$\frac{4\pi}{3}$$) + z² sin($$\frac{6\pi}{3}$$) + ... + z^k sin(k$$\frac{2\pi}{3}$$) + ...

Homework Equations

$$\sum$$ n =1 to $$\infty$$ (e^2pi*i/3z)ⁿ = 1 + $$\sum$$ n =1 to $$\infty$$ (e^2pi*i/3z)ⁿ

The Attempt at a Solution

 

[vela]

What's the question and where's your attempt at solving it?

 

[FanofAFan]

Not sure where to begin, thought I'd get some type of hint, but I figured let z = x +iy and change the series to sin(2pi/3) + x sin(4pi/3) + iy sin(4pi/3) + x² sin(6pi/3) - y² sin(6pi/3) + 2ixy sin(6pi/3) +...+ and maybe separate the series in real and imaginary parts, I'm completely off

 

[vela]

Do you know what the sum of the series you listed in the relevant equations section is?

 

[FanofAFan]

Si
it is 1+e^2pi*i/3z + e^4pi*i/3z² + e^6pi*i/3z³ + .. + e^k(2pi*i/3)z^k

 

[vela]

I meant, do you know what it sums to? Hint: It's a geometric series.

 

[FanofAFan]

Is it Maclaurin series for exponental... so 1 + e^(2i pi/3) z = cos(2 pi/3) + i sin(2 pi/3) z + and so on... right?

 

[vela]

No. If you have the geometric series

$$\sum_{n=0}^\infty r^n$$

with |r|<1, what does it sum to?