Uniform and absolute convergence of $\sum \frac{2^n}{n^2+1} x^{2n}$

Question

Find the radius of converence of the series

$$\sum \frac{2^n}{n^2+1} x^{2n}$$

and analyze the absolute and/or uniform convergence

My attempt

1. The convergence radius of this serie is $\frac{1}{\sqrt{2}} ( x^{2n}=(x^2)^n)$

2. If $x=\frac{1}{\sqrt{2}},$ the series converges, since it is the sequence $\sum\frac{1}{n^2+1}$

3. If $x=\frac{-1}{n}$ also converges, applying the Dirichet criteria for alternating series

My doubt is when we talk about the uniform convergence:

Since the serie $\sum \frac{2^n}{n^2+1} (\frac{1}{\sqrt{2}})^{2n}$ converges and

$$||{f_n}||_{\infty} = \frac{1}{n^2+1}$$

by the M-test the series must converges uniformly on $[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}], $ correct?

Thanks in advance!