Pointwise and uniform convergence of power series

Question

I want to check the pointwise and uniform convergence of $$\sum_{n=1}^{+\infty}\frac{x^ne^{-n}}{\sqrt{n}}$$

For the pointwise convergence do we check the limit of the sequence?

I mean the following: $$a_n=\frac{x^ne^{-n}}{\sqrt{n}} \rightarrow \lim_{n\rightarrow +\infty}a_n=\lim_{n\rightarrow +\infty}\frac{x^ne^{-n}}{\sqrt{n}}=\lim_{n\rightarrow +\infty}\frac{x^n}{\sqrt{n}e^{n}}=0$$ Therefore the series converges pointwise to $0$.

Is that correct?

And for the uniform convergence do we check also the sequence?

Or do we have to do something else for the series?

Answer 1

By well known Cauchy–Hadamard theorem for power series $\sum\limits_{n=0}^{\infty}(z-z_0)^nc_n$ we have, that so called convergence radius $\frac{1}{R}=\lim\limits_{n \to \infty}\sup\sqrt[n]{|c_n|}$. In our case

$$\sqrt[n]{\frac{e^{-n}}{\sqrt{n}}} \to \frac{1}{e}=\frac{1}{R}$$

So we have pointwise convergence for $|x|<e$. In right border point we have divergence as for $\frac{1}{\sqrt{n}}$ and for left convergence $\frac{(-1)^n}{\sqrt{n}}$.

As it is known, uniform convergence we have on each closed segment within convergence interval.