Non-zero terms in a series

Question

Suppose I know that |$\sum_{k=0}^\infty a_kz^k$| converges to $0$ as $|z| \to \infty$, and $\sum_{k=0}^\infty a_kz^k$ converges absolutely for every $|z| >1$ ($z$ is complex). Is it true that all the $a_k$ are equal to zero. This seems to be so but I am unable to prove it.

Basically, if there is a lower bound for |$\sum_{k=0}^\infty a_kz^k$| in terms of $|z|$ then this might be used to show that convergence to $0$ fails. However, I have been unsuccessful with this approach. For example, splitting the $a_k$ into positive and negative parts in the hope of making the absolute value behave better did not work.

Jonas Dahlbæk · Accepted Answer · 2014-09-04T10:17:41.287

3

The Maximum Modulus Principle implies that what you have is a bounded entire function. Liouville's Theorem then implies that the function is constant, and considering the boundary behaviour 'at infinity' you then see that the function vanishes everywhere.

edited Sep 04 '14 at 10:17

answered Sep 04 '14 at 10:12

Jonas Dahlbæk

4,776

Ah, I have missed that $\sum_{k=0}^\infty a_kz^k$ converges absolutely only in $|z| > 1$. This is not an entire function then. – jpv Sep 04 '14 at 10:23
By the Root Test, if your series converges absolutely for $z=2$, then it also converges absolutely for $|z|<2$. – Jonas Dahlbæk Sep 04 '14 at 10:27

Non-zero terms in a series

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