Uniform Convergence of $\sum_{n=1}^\infty \frac{1}{(n+x)^2}$

Question

Does the sum $$\sum_{n=1}^\infty \frac{1}{(n+x)^2}$$ converge uniformly for $x\in \mathbb{R}$?

Answer 1

For $x\in \mathbb{R}$, and $x > 0$. Note that $$\sum_{n=1}^{\infty}\frac{1}{(n+x)^2} \leq \sum_{n=1}^{\infty}\frac{1}{n^2}$$ Since $$\lim_{N\to \infty}\sum_{n=1}^{N}\frac{1}{n^2} \to 0$$ Then by the Weierstrass M-test $\sum_{n=1}^{\infty}\frac{1}{(n+x)^2}$ is uniformly convergent.