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I am not a pure mathematician, so I would appreciate some help from people who have done analysis! Can we have some function which is analytic (which I believe just means expressible as a power series) but the term by term integral does not converge or somehow goes wrong? If so, what are the requirements for the function so that this does not happen?

Also, suppose we don't do the integration term by term, then would it always be true that an integral is well-defined/exists where the power series of the function "works"?

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A power series $\sum_i a_ix^i$ has radius of convergence at least $r$ whenever $a_nr^n$ has at most sub-exponential growth as $n\to\infty$, that is if $a_nr^n=o(\lambda^n)$ for every $\lambda>1$. Taking a formal primitive gives $\sum_{i>0} \frac{a_{i-1}}ix^i$, and it is easy to see that $\frac{a_{n-1}}nr^n=o(\lambda^n)$ if and only if $a_nr^n=o(\lambda^n)$, so the radius of convergence is unchanged by taking a primitive.

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