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## Homework Statement

##P(z) = 1 - \frac{z}{2} + \frac{z^2}{4} - \frac{z^3}{8} + ... ##

Determine if the series is convergent or divergent if ## |z| = 2 ##, where, ## z## is a complex number.

## Homework Equations

##1+r+r^2+r^3+...+r^{N-1}=\frac{1-r^N}{1-r}##

## The Attempt at a Solution

Let, ##z = 2 exp (i \theta)##[/B]

For the first ##N## terms, the summation is,

##P_N (z) = \frac{1-(-1)^N exp(iN\theta)}{1+exp(i \theta)}=\frac{1-(-1)^N \cos (N\theta) - i (-1)^N \sin (N \theta)}{1 + exp (i \theta)}##

As ## N \rightarrow \infty##, ##\cos (N \theta)## and ## \sin (N \theta)## do not converge to a particular value.

So, I conclude that the series is not convergent for ## |z|=2##

But the answer says that the series converges to ##\frac{1}{1+\frac{z}{2}}##

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