Evaluate if a series converges to a continuous function

Question

I am trying to show that a function defined by this series $$ \sum_{n=1}^{\infty} \frac{1}{x^2+n^2}$$ is a continuous over $\mathbb{R}$.

I am not sure what to do but I think if it is possible to show that the series converges uniformly then I will be able to say that it converge to a function $f$ represents the sum which is also a continuous function using the uniform convergence properties.

I am looking for a help for my current difficulty in proving that the series converge uniformly to a function.

Answer 1

Use the Weierstrass $M$-test: for each $x\in\mathbb R$ and each $n\in\mathbb N$, $\dfrac1{x^2+n^2}\leqslant\dfrac1{n^2}$. Since the series $\displaystyle\sum_{n=1}^\infty\frac1{n^2}$ converges, your series converges uniformly.