finding if a series converges uniformly

Question

If I have this series $$\sum_{k=1}^{\infty} \frac{k^2}{\sqrt{k!}} (x^k+x^{-k})$$ then how can I find out if it converges uniformly or not when $\frac{1}{2}\leq|x|\leq 2$? My first thought was to use Weierstrass M-test, but im not sure if that would really work out.

Answer 1

If $\frac12\leqslant \lvert x\rvert\leqslant2$, then $\lvert x\rvert^k,\lvert x\rvert^{-k}\leqslant2^k$ and therefore$$\frac{k^2}{\sqrt{k!}}\lvert x^k+x^{-k}\rvert\leqslant\frac{k^22^{k+1}}{\sqrt{k!}}.$$But$$\frac{\frac{(k+1)^22^{k+2}}{\sqrt{(k+1)!}}}{\frac{k^22^{k+1}}{\sqrt{k!}}}=\left(1+\frac1k\right)^2\frac2{\sqrt{k+1}}\leqslant\frac8{\sqrt{k+1}}\to_{k\to\infty}0.$$Therefore, yes, your series converges uniformly by the Weierstrass $M$-test.