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Unformatted text preview: So all three series converge for all complex numbers z with z  < R and diverge for all
complex numbers with z  > R. What if z  = R?
∞
zn
We’ll start with the series n=0 R . Then we can compute exactly the partial sum 1 − z n+1
n R
if z = R
zm
z
=
Fn (z ) =
(2)
1− R
R m=0
n+1
if z = R
1
z
1− R As expected, if z  < R this converges to
for z  > R, because z
R n+1 = z n+1
R as n → ∞. Also as expected, this diverges → ∞. I claim that this also diverges whenever z  = R. For z = R, it is obvious because n + 1 → ∞. For z  = R with z = R,
does not blow up as n → ∞, but it cannot converge either, because
z n+2
R − z n+1
R = z n+1
R z...
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This document was uploaded on 02/24/2014 for the course MATH 321 at University of British Columbia.
 Winter '08
 JoelFeldman
 Math

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