# Is there an entire function with a conditionally convergent power series?

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
No. What happens when $z$ is taken to be a unit? – user18063 Jan 6 '12 at 3:51

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.