I am just solving many question on series of uniform convergence in order to get more intuition and experience with this stuff so I wanted to know what do you guys think of my argument. Its easy for like sequence but series sometimes is hard to deal with I am kinda confused on series for uniform convergence if someone could clarify this issue that would be great.
1)$\sum 1/(nx)^2$ for $x \in (0,1]$ so this will diverge and the reason for that we just substitute x = 1/n so we get $\sum 1$ so this will won't even converge point-wise therefore we and will diverge so since $\lim \sum 1$ is infinity Hence won't converge uniformly.
2)$\sum x^2/n^2$ for $x \in [5,\infty]$ I think diverge if we choose for x > n we choose x = 1/n and therefore will diverge by same argument I had above.
3)$\sum 1/(1 + n^2x^2)$ for $x \in (0,1]$ This will diverge if we choose x = 1/n and use the same argument as above.