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Does there exist an entire (holomorphic on all of $\mathbb C$) function $f(z) =\displaystyle\sum_{n=0} ^\infty a_n z^n$ such that $\displaystyle\sum_{n=1} ^\infty |a_n| = \infty$? If not, how can one prove that there is no such function?

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A power series converges absolutely in the interior of its disc of convergence. Since $1$ is in the interior of the disc of convergence of the Taylor series of an entire function, the answer to your question is no. –  Mariano Suárez-Alvarez Jan 6 '12 at 3:49
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No. What happens when $z$ is taken to be a unit? –  user18063 Jan 6 '12 at 3:51

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No. By the root test, $\sum\limits_{n=1}^\infty |a_n|=\infty$ implies $\limsup\limits_{n\to\infty}\sqrt[n]{|a_n|}\geq 1$. Then $\limsup\limits_{n\to\infty}\sqrt[n]{|a_n|2^n}\geq 2$, which by the root test implies $\sum\limits_{n=1}^\infty a_n2^n$ diverges.

In general, a power series converges absolutely everywhere within its disk of convergence, and may converge either conditionally or absolutely (or not at all) on the boundary. For entire functions, the convergence is absolute everywhere. A basic reference is the Wikipedia article on the radius of convergence.

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