Show That $$\sum_{n=1}^\infty{1\over x ^2+n^2} \sim \frac1x$$ as $x\to \infty.$
It is enough to show that $\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac x{n^2+x^2}$ exists and is positive
$$=\lim_{x\to\infty}\lim_{k\to\infty}{x\over x^2}\sum_{n=1}^{kx}{1\over 1+(n/x)^2}$$ $$=\lim_{k\to\infty}\lim_{x\to\infty}{1\over x}\sum_{n=1}^{kx}{1\over 1+(n/x)^2}$$ $$=\lim_{k\to\infty}\int_0^k{1\over1+n^2}dn$$ $$=\int_0^\infty{1\over1+n^2}dn=\pi/2$$
According to Desmos, my calculations are correct but I'm worried about the rigour of the proof. How Do I justify my limit swaps and conversion to Riemann sum
Let $S(x)=\sum_{n=1}^{\infty}\dfrac{1}{x^2+n^2}.$ Note that for a fixed $x,$ $\dfrac{1}{x^2+t^2}$ is decreasing in $t.$ Therefore
$$\int_2^\infty \frac{dt}{x^2+t^2} \le S(x) \le \int_1^\infty \frac{dt}{x^2+t^2}.$$
This gives
$$\frac{1}{x}(\pi/2-\arctan(2/x))\le S(x) \le \frac{1}{x}(\pi/2-\arctan(1/x)).$$
Your result follows.