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Pyroadept
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Homework Statement
Find the radius of convergence of the series:
∞
∑ n^-1.z^n
n=1
Use the following lemma:
∞ ∞
If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also
n=1 n=1
converges.
The contrapositive is also true.
Homework Equations
The Attempt at a Solution
Hi, here's what I've done:
Let r be the radius of convergence. The series converges when |z-w| < r and diverges for |z-w|> r.
In this question, z = z, w = 0.
Use the ratio test for convergence:
lim |a_n+1 / a_n| = lim |zn / n+1 |
n->∞
= |z|
Thus the series converges when |z| < 1
diverges when |z| > 1
So let |z - 0| = |z_2 - w|
Then, by the lemma, all |z_1 - 0| converges, where |z_1 - 0| < |z - 0|
and the converse is true for a diverging series.
Thus the radius of convergence = |z|
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I think I've covered everything, but I don't think I've made very satisfactory use of the lemma. Can anyone please point me in the right direction?
Thanks for any help.