(adsbygoogle = window.adsbygoogle || []).push({}); Using De Moivre's theorem to prove the sum of a series

1. The problem statement, all variables and given/known data

Write down an expression in terms of z and N for the sum of the series:

[tex]\sum_{n=1}^N 2^{-n} z^n[/tex]

Use De Moivre's theorem to deduce that

[tex]\sum_{n=1}^{10} 2^{-n} \sin(\frac{1}{10}n\pi)[/tex] = [tex]\frac{1025\sin(\frac{1}{10}\pi)}{2560-2048\cos(\frac{1}{10}\pi)}[/tex]

2. Relevant equations

[tex] e^{in\theta}=(\cos{\theta}+i\sin{\theta})^n = \cos{n\theta}+i\sin{n\theta}[/tex]

3. The attempt at a solution

To find a sum for the series it is a GP with first term,[tex]a=2^{-1}z[/tex] common ration,[tex]r=2^{-1}z[/tex]

then [tex]S_N = \frac{2^{-1}z(1-(2^{-1}z)^{N})}{1-2^{-1}z}[/tex]

For the second part I was thinking to just replace [tex]z^n[/tex] with [tex]\sin(\frac{1}{10}n\pi)[/tex] would that work?

(NOTE:Also, even though I think I typed the LATEX thing correctly it doesn't display what i actually typed when i previewed the post, so if something looks weird please check if I typed it correctly,such as

\frac{1025\sin(\frac{1}{10}\pi)}{2560-2048\cos(\frac{1}{10}\pi) appears as [tex]a^3<-9b-3c-3[/tex]

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# Homework Help: De Moivre's theorem

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