# Connection between properties of taylor series and the function

Assuming I have a function $f(x)$ which at least for some $-R<x<R$ can be expanded in taylor series

$$f(x) = \sum_{n=0}^{\infty}c_n \frac{x^n}{n!}$$

are there any known connections between properties of the series $\{c_n\}_{n=0}^\infty$ and properties of the function itself?

I have not seen any in textbooks and it seems odd given the importance of taylor series.

The kind of connection of properties I'm talking about would be: if $\{c_n\}_{n=0}^\infty$ is periodic then the function is periodic (I know this one is incorrect, but just to illustrate).

• "If $(c_n)_n$ only has finitely many non-zero terms, then $f$ is a polynomial?" (on $(a,b)$) – Clement C. Dec 28 '15 at 13:41
• If $c_n=0$ for all odd $n$ ($1,3,5,\ldots$), then $f$ is an even function (that is, $f(-x)=f(x)$. Similarly, if $c_n=0$ for all even $n$ ($0,2,4,\cdots$), then $f$ is an odd function (that is, $f(-x)=-f(x)$). – MPW Dec 28 '15 at 13:48
• When you say "can be expanded in taylor series for some $x$", do you mean "in a neighborhood of some $x$" ? – Gabriel Romon Dec 28 '15 at 13:48
• @LeGrandDODOM: As stated, the series is expanded about $x=0$ and converges at least in the interval $(a,b)$. Of course, one may assume this interval of convergence is of the form $(-r,r)$ since the center is $0$. – MPW Dec 28 '15 at 13:51
• @LeGrandDODOM yes, I've corrected the post, thanks – xaxa Dec 28 '15 at 14:34