- #1
arpon
- 235
- 16
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}}##
Last edited: