Uniform convergence of $f_n(x) = \frac{x^ne^{-x}}{(2n)!}$

Question

I came across this question while studying for my qualifying exams and it was grouped together with problems involving the Weierstrass M-test.

Let $f_n(x) = \frac{x^ne^{-x}}{(2n)!}$. Does $(f_n(x))$ converge uniformly on $[0,\infty)$?

I don't think I can apply the M-test since we are not looking at the series $\sum f_n(x)$, and I can't apply Dini's theorem since $[0,\infty)$ is not compact. Hints are greatly appreciated.

Answer 1

hint

For $x\ge 0$,

$$f'_n(x)=\frac{1}{(2n)!}x^{n-1}e^{-x}(n-x)$$

So, $$M_n=\max_{[0,+\infty)}\{|f_n(x)-0|\}=f_n(n)$$

$$=\frac{n^ne^{-n}}{(2n)!}$$

Now $$\lim_{n\to+\infty}\frac{M_{n+1}}{M_n}=$$ $$\lim_{n\to+\infty}(1+\frac 1n)^n\frac{n+1}{(2n+1)(2n+2)e}=0$$ $$\implies \sum M_n \text{ converges}$$ $$\implies \lim_{n\to+\infty}M_n=0$$ $$\implies \text{ the convergence is uniform}$$