# Pointwise and uniform convergence of power series

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?

• It is not true that if $a_n \to 0$, then $\sum_n a_n$ converges, so your test is insufficient. Aug 22 '20 at 13:20
• Uniform convergence mean set for $x$. Which one you have here? Aug 22 '20 at 13:20
• It might help if you expand on what the definition of "pointwise and uniform convergence" is in this context. When one asks where a power series over $x$ converges pointwise, the question is: for which values of $x$ does it hold that the series converges? Aug 22 '20 at 13:23

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 $$|x|. In right border point we have divergence as for $$\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.

• @Mary Star. By the way, I wrote for you proof which you asked about on math.stackexchange.com/questions/3797499/… - do you see it? Aug 22 '20 at 14:02
• For the n-th root test we don't consider the $x^n$-term,right? Aug 24 '20 at 5:15
• @Mary Star. Right. Link and formulation added to answer. If/when you'll have more question for this or other answers, feel free to ask. Aug 24 '20 at 7:46
• @zkutch the ratio accepted answers / answers recently received of this user is tiny! 2 in the last 28 i.e. about 7%. (28 is what's on one page of their account). Seems to be better towards the older of their questions (36 pages!) Aug 24 '20 at 11:17
• @Calvin Khor. Thanks. I like that name "Mary".. Aug 24 '20 at 12:13