continuity of $f(x)=\sum_1^{\infty}\left(x+\frac{1}{n^2}\right)^n$ verification

Question

I wish to check for which $x$'s the function:

$$f(x)=\sum_1^{\infty}\left(x+\frac{1}{n^2}\right)^n$$

is continuous.

I'm not sure I solved it the right way, please let me know if I made mistakes.

Here is what I did:

I first check for uniform convergence:

for $x\le-1$ or $x\ge 1$ we get that $\lim_{n\rightarrow \infty}f_n(x)\ne 0$ so it is not an option.

for $-1\lt x \lt 1$ I used the root test:

$$\lim_{n\rightarrow \infty}\left|\sqrt[n]{\left(x+\frac{1}{n^2}\right)^n}\right|=|x|$$

so for $-1\lt x \lt 1$ the series uniformly converges, and since each $f_n$ is continuous, $f$ is continuous there.

Any corrections will be very helpful.

Answer 1

Hint: The series does not converge uniformly on $(-1,1).$ You can see this by noting $\sup_{(-1,1)} |f_n| \ge 1$ for each $n.$ However you may be able to show the series converges uniformly on $[-a,a]$ for each $a, 0<a<1.$ That would prove $f$ is continuous on each $[-a,a],$ and that will be enough to give the conclusion you claimed.