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Unformatted text preview: What if
= n=0 n which diverges. So
|z | = R? Well, if z = R, then the series is n=0 n R
that leaves |z | = R but with z = R. This is where the Dirichlet test comes in handy. Fix
any ε > 0 and set X=
fn ( z ) = z∈C
R Fn (z ) =
February 3, 2008 m=0
n |z | = R, |z − R| ≥ ε X n as in (2) |z − R | = ε
|z | = R
The Dirichlet Test 2 For z ∈ X n+1 z
Fn (z ) =
1− R n+1 1+ z
|R − z |
R so that the hypotheses of the Dirichlet test are satisﬁed and the series converges uniformly
on X . We conclude that n=0 n R converges for |z | < R and for |z | = R, z = R and
diverges for |z | > R and for z = R. February 3, 2008 The Dirichlet Test 3...
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This document was uploaded on 02/24/2014 for the course MATH 321 at University of British Columbia.
- Winter '08