On the Series $\sum_{n=0}^\infty t^nf^{(n)}(t) (-1)^n/n!$

TL;DR

This paper investigates the convergence properties of a specific infinite series involving derivatives of functions, establishing conditions under which it converges to the function's value at zero for analytic and certain non-analytic functions.

Contribution

It characterizes the convergence behavior of the series for both analytic and non-analytic functions, revealing new insights into its uniform, absolute, and pointwise convergence.

Findings

Series converges to f(0) for analytic functions.

Series diverges for some nowhere analytic functions.

Series converges to f(0) on a dense subset for certain nowhere analytic functions.

Abstract

We study the series $\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}t^nf^{(n)}(t)$. We show that for analytic functions this series is uniformly and absolutely convergent to the constant $f(0)$. We show that there are nowhere analytic functions for them the series is divergent for all $t$ and also there are nowhere analytic functions for them the series is convergent to $f(0)$ at least for $t$ in a dense subset of $\R$.