I am asked to show that the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2+x^2}, x \in \mathbb{R}$$ converges uniformly in $\mathbb{R}$ to a $s(x)$ and that $s(x)$ is continuous.
That's what I have tried:
$$n^2+x^2 \geq n^2 \Rightarrow \frac{1}{n^2+x^2} \leq \frac{1}{n^2} \Rightarrow \sum_{n=1}^{\infty} \frac{1}{n^2+x^2} \leq \sum_{n=1}^\infty\frac{1}{n^2} < +\infty$$
So,from the Weierstrass criterion, $\sum_{n=1}^{\infty} \frac{1}{n^2+x^2} \overset{ uniformly}{=} s(x)$.
As $\frac{1}{n^2+x^2}$ continuous and the convergence is uniform, $s(x)$ is also continuous in $\mathbb{R}$.
Could you tell me if it is right or if I have done something wrong?
