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?
Is there an entire function with a conditionally convergent power series?
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sequences-and-series
complex-analysis
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2No. What happens when $z$ is taken to be a unit? – 2012-01-06
1 Answers
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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.