Prove the series $\sum_{n=1}^{\infty} \frac{x^{2n-1}}{(2n-1)!}$ converges uniformly on any bounded interval of $\mathbb{R}$.

Question

Prove the series $\sum_{n=1}^{\infty} \frac{x^{2n-1}}{(2n-1)!}$ converges uniformly on any bounded interval of $\mathbb{R}$. I was thinking about Weierstrass M-test but I got stuck. Thank you so much for your time and help.

Answer 1

Let $I$ be a bounded intervall. Then there is $c>0$ such that $|x| \le c$ for all $x \in I$. It follows that

$|\frac{x^{2n-1}}{(2n-1)!}| \le \frac{c^{2n-1}}{(2n-1)!}$ for all $x \in I$ and all $n$.

Since $\sum_{n=1}^{\infty} \frac{c^{2n-1}}{(2n-1)!}$ is convergent, the Weierstraß M - test shows that $\sum_{n=1}^{\infty} \frac{x^{2n-1}}{(2n-1)!}$ converges uniformly on $I$.