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Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$

I have to prove that : Given a power series $\sum_n a_n x^n$ diverges for $x = x'$ then it diverges for every $x$ such that $| x | > |x'|$ . Please suggest how to proceed.

I am trying to use the First comparison test, knowing the series $\sum_n a_n x'^n$ is not convergent which implies that it is not absolutely convergent.

Then comparing the $n$'th terms of the series we have

$ | a_n x'^n | < | a_n x ^n | $

Since the series on the left is not convergent, we have that the series on the right is not convergent either.

But I am getting the not absolute convergence of the series by this approach...Please help !

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    @JyrkiLahtonen: Thanks, will do so.2012-04-06

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Try to prove it by contradiction. Have you proven that

Theorem: If a power series converges for a number $z=z_0\neq0$ then it will converge absolutely for any $z$ such that $|z|<|z_0|$.

Suppose that a series diverges for $z=z_0$. Then assume it converges for $z=z_1$ such taht $|z_1|>|z_0|$. Then what would happen to the series for $z_0$ given the above theorem is true?