Discuss the convergence and the uniform convergence of a series

Question

So the series to be discussed is $\sum (nx)^{-2}$ $\ (x\ne0)$.
And I think I can show that the series is pointwise convergent in its domain by comparison test with $\frac{1}{n^2}$.
But I'm not sure how to show that it's not uniformly convergent(I think). What I currently know is that Weierstrass M-test cannot be applied. And I feel that I have to use the Cauchy Criterion? But I'm not sure how to write the proof.

Answer 1

Assume it were uniformly convergent. Then there exists some positive integer $N$ such that for all $x\ne 0$ we have $\left|\displaystyle\sum_{k=1}^{N}\dfrac{1}{k^{2}x^{2}}-\displaystyle\sum\dfrac{1}{n^{2}x^{2}}\right|<1$, then $\displaystyle\dfrac{1}{x^{2}}\sum_{k\geq N+1}\dfrac{1}{k^{2}}=\displaystyle\sum_{k\geq N+1}\dfrac{1}{k^{2}x^{2}}<1$, so $0<\displaystyle\sum_{k\geq N+1}\dfrac{1}{k^{2}}<x^{2}$ for all $x\ne 0$. Take $x\downarrow 0$ we get $\displaystyle\sum_{k\geq N+1}\dfrac{1}{k^{2}}=0$, a contradiction.