My colleague Tom Liggett recently posed to me the following problem about power series in one real variable ${x}$. Observe that the power series $\displaystyle \sum_{n=0}^\infty (-1)^n\frac{x^n}{n!}$

has very rapidly decaying coefficients (of order ${O(1/n!)}$), leading to an infinite radius of convergence; also, as the series converges to ${e^{-x}}$, the series decays very rapidly as ${x}$ approaches ${+\infty}$. The problem is whether this is essentially the only example of this type. More precisely:

Problem 1 Let ${a_0, a_1, \dots}$ be a bounded sequence of real numbers, and suppose that the power series $\displaystyle f(x) := \sum_{n=0}^\infty a_n\frac{x^n}{n!}$

(which has an infinite radius of convergence) decays like ${O(e^{-x})}$ as ${x \rightarrow +\infty}$, in the sense that the function ${e^x f(x)}$ remains bounded as ${x \rightarrow +\infty}$. Must the sequence ${a_n}$ be of the form ${a_n = C (-1)^n}$ for some constant ${C}$?

As it turns out, the problem has a very nice solution using complex analysis methods, which by coincidence I happen to be teaching right now. I am therefore posing as a challenge to my complex analysis students and to other readers of this blog to answer the above problem by complex methods; feel free to post solutions in the comments below (and in particular, if you don’t want to be spoiled, you should probably refrain from reading the comments). In fact, the only way I know how to solve this problem currently is by complex methods; I would be interested in seeing a purely real-variable solution that is not simply a thinly disguised version of a complex-variable argument.

(To be fair to my students, the complex variable argument does require one additional tool that is not directly covered in my notes. That tool can be found here.)