Continuity of $f(x)= \sum _ {n=0} ^{\infty } \left ( \frac{x^n}{n!} \right )^2$ Recently, I've been asked to prove that this function is continuous on $\mathbb{R}$
$$f(x)= \sum _ {n=0} ^{\infty } \left ( \frac{x^n}{n!} \right )^2$$
I do recognize its very similar to the numerical series that defines $e$, yet I am working here with sequences and series of functions, so I don't know wheter this is helpful or not. Also, I think it could be helpful to try to use the uniform limit theorem and Weierstrass' M test.
I'm blank, and I would appreciate some little advices about how I can start working on the proof and what I should aim to in order to get the proof done and what thorems should I give a look. Thanks for your attention!!!
 A: For $A\in \Bbb N$ let $f_A=\sum_{n=0}^A(x^n/n!)^2.$ For $M>0$ let $\|f-f_A\|_M=\sup \{|f(x)-f_A(x)|: |x|<M\}.$
Then for any $M>0$ we have $\lim_{A\to \infty}\|f-f_A\|_M=0.$
Because for each $|x|<M$ we have $$|f(x)-f_A(x)|=|\sum_{n>A}(x^n/n!)^2|\le \sum_{n>A}(|x|^n/n!)^2<$$ $$<\sum_{n>A}(M^n/n!)^2=^{def} g_A(M)$$ and for a fixed $M$ we have $\lim_{A\to \infty}g_A(M)=0.$
In other words $f_A\to f$ uniformly (as $A\to \infty$) on any $(-M,M).$
Theorem: If a sequence $(f_A)_{A\in \Bbb N}$ of continuous real functions converges uniformly to $f$ on every bounded subset of $\Bbb R$ then $f$ is continuous.
Proof: Given $x\in \Bbb R$ and $\epsilon >0,$ we can find $\delta>0$ such that $|y-x|<\delta\implies |f(y)-f(x)|<\epsilon$ as follows:
Take $M>|x|.$ Take some big enough $A\in \Bbb N$ such that $\sup \{|f(y)-f_A(y)|: |y|<M\}<\epsilon/3.$ Now $f_A$ is continuous at $x$ so take some $\delta>0$ small enough that $(-\delta+x,\delta+x)\subset (-M,M)$ and such that $|y-x|<\delta\implies  |f_A(y)-f_A(x)|<\epsilon/3.$
So if $|y-x|<\delta$ then $$|f(y)-f(x)|\le |f(y)-f_A(y)|+|f_A(y)-f_A(x)|+|f_A(x)-f(x)|<$$ $$<\epsilon/3+\epsilon/3+\epsilon/3.$$
This theorem applies verbatim to $\Bbb C$ if we replace $(-M,M)$ with $\{z\in \Bbb C:|z|<M\}$ and replace $(\delta+x,\delta+x)$ with $\{z: |z-x|<\delta\}.$
You should also study the proof of the Cauchy-Hadamard Radius Formula for more insight about power series.
