Calculus 1
Criterion for Boundedness of Power Series
Introduction & Problem Statement # Power series constitute one of the most ubiquitous objects in analysis. A power series $\sum_{n=0}^{\infty}a_n x^n$ with infinite radius of convergence defines a real-entire function $f:\mathbb{R}\to\mathbb{R}$. Whereas the question of convergence is completely settled by Cauchy–Hadamard theory, the question of boundedness of the sum function is far subtler and, as of this writing, remains open. Question 1 (Rüdinger, 2009) Let $(a_n) _{n\ge 0}$ be a sequence of real numbers such that the power series $\sum _{n=0}^{\infty}a_n x^n$ converges for every $x\in\mathbb{R}$, thereby defining a smooth function $f:\mathbb{R}\to\mathbb{R}$. Give a necessary and sufficient criterion on $(a_n)$ for $f$ to be bounded on $\mathbb{R}$.