Convergence of Series with Ratio Test

Question

Prove $$\sum_{n=0}^{\infty}\frac{1}{1+(x-n)^{2}}$$ converges for $x \in \mathbb{R}$.

I am having difficulty solving this using the ratio test.

Answer 1

$$\sum_{n=0}^{\infty}\dfrac{1}{1+(x-n)^{2}} =\sum_{n=0}^{k}\dfrac{1}{1+(x-n)^{2}} +$$

$$\sum_{n=k+1}^{\infty}\dfrac{1}{1+(x-n)^{2}} $$ where $k$ is an integer larger than $x+1$.

The first sum is no problem.

The second one $$\sum_{n=k+1}^{\infty}\dfrac{1}{1+(x-n)^{2}}\le \sum_{n=1}^{\infty}\dfrac{1}{1+n^{2}}$$

Which is convergent.