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geoffrey159
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Homework Statement
Given the power serie ##\sum_{n\ge 0} a_n z^n##, with radius of convergence ##R##, if there exists a complex number ##z_0## such that the the serie is semi-convergent at ##z_0##, show that ##R = |z_0|##.
Homework Equations
The Attempt at a Solution
Firstly, since ##\sum_{n\ge 0} a_n z_0^n## is not absolutely convergent, then ## R \le |z_0|##.
Secondly, ##\sum_{n\ge 0} a_n z_0^n## is convergent so ##(a_n z_0^n)_n## is bounded as it tends to zero, so ## |z_0| \le R##.
So ##R = |z_0|##.
Is it correct ?
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