Why does the convergence radius even exist for power series

Question

WHY do power series diverge outside of the convergence radius? Or: Why does the convergence radius for power series even exist?

Answer 1

It's because of the following not entirely obvious property: if a power series $\sum_{n \geq 0} a_nx^n$ converges at a number $x_0$ then it converges at all $x$ with $|x| < |x_0|$, so it converges on $(-x_0,x_0]$ (and maybe not $-x_0$). So it is in the nature of power series to converge on intervals, with or without some of the endpoints.

For example, suppose we know $\sum_{n \geq 0} a_nx^n$ converges at $x = 1$. Then it converges on $(-1,1]$. Next, suppose we find out it also converges at $x = -2$. Then it converges on $[-2,2)$. The interval of converges might get bigger, but certainly not smaller.

If a power series does not converge at some number $c$ then it does not converge at any $x$ with $|x| > |c|$ (because if it converged at such $x$ it would converge at all smaller numbers in absolute value, and that includes $c$ where we're told it doesn't converge).

Ultimately what this means is the domain of convergence, if it is not all of $\mathbf R$, will be some symmetric interval of the form $(-R,R)$ with or without one of the endpoints $\pm R$. This can be made precise using the notion of a supremum: if $S$ is a nonempty subset of $\mathbf R$ such that whenever a number $x_0$ is in $S$ we have $x \in S$ for all $x$ with $|x| < |x_0|$, and $S$ is not all of $\mathbf R$, then $(-R,R) \subset S \subset [-R,R]$ where $R = \sup_{x \in S} |x|$.

As a contrast to this, consider another important infinite series expansion: the Fourier series $a_0 + \sum_{n \geq 1} (a_n\cos(nx) + b_n\sin(nx))$. If you are told this series converges at a number $x_0$, you do not learn anything about where else the series converges. Convergence of Fourier series is a very delicate issue. There is no simple description of the set of numbers at which a general Fourier series converges, although there are some sufficient conditions for convergence of suitably well-behaved periodic functions. The many subtle problems connected with understanding convergence of Fourier series led to a huge amount of mathematics in the 19th and early 20th centuries: the meaning of convergence, the definition of a function, Riemann's definition of his integral, measure theory, Cantor's work on set theory, and so on. That is why David Bressoud chose to begin his historically-oriented account of real analysis (A Radical Approach to Real Analysis) by starting with a chapter on Fourier series as a "crisis in mathematics". Srivastava's article here explains how Cantor's work on Fourier series (or "trigonometric series" in the old language) inspired his discoveries in set theory.