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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.
The third series,
, converges for all |z | ≤ R, by comparison with
n=0 n2 R
n=0 n2 . As the series has radius of convergence R, it converges if and only if |z | ≤ R.
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....
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This document was uploaded on 02/24/2014 for the course MATH 321 at University of British Columbia.
- Winter '08