This preview shows page 1. Sign up to view the full content.
Unformatted text preview: R −1 = z
R z n+1
R −1 n ∞ z
is independent of n. So the geometric series n=0 R , which has radius of convergence
R, converges if and only if z  < R.
∞
1 zn
The third series,
, converges for all z  ≤ R, by comparison with
n=0 n2 R
∞
1
n=0 n2 . As the series has radius of convergence R, it converges if and only if z  ≤ R.
∞
1zn
The middle series n=0 n R
has a more interesting domain of convergence. Of
course the radius of convergence is exactly R, so the series converges for all complex
numbers z with z  < R and diverges for all complex numbers with z  > R....
View
Full
Document
This document was uploaded on 02/24/2014 for the course MATH 321 at University of British Columbia.
 Winter '08
 JoelFeldman
 Math

Click to edit the document details