uniform convergence of a function (continuous or differentiable or both?)

Question

I have a function $S$:

$$ S(x) = \sum_{n=1}^\infty \frac1{x+n^2} \\ \text{for} \ x \ge 0 $$

I need to determine if $S$ is continuous or differentiable or both.

Answer 1

Because $$ \frac1{x+n^2}\leq\frac1{n^2} $$ the partial sums converge uniformly from Weierstrass test, hence $S(x)$ is continuous.

Because $$ \left(\frac1{x+n^2}\right)'=-\frac1{(x+n^2)^2} $$ we can again use Weiestrass test to the derivatives, obtaining that $S(x)$ is differentiable.