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?
Best Answer
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.