Question

uniform convergence

Answer 1

@bibby

Answer 2

I'm thinking you can use Dirichlet's test (see here: https://proofwiki.org/wiki/Dirichlet%27s_Test_for_Uniform_Convergence ) with
\[\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n+x^2}=\sum_{n=1}^\infty \underbrace{\frac{(-1)^{n-1}}{n}}_{a_n}\underbrace{\frac{1}{1+\frac{x^2}{n}}}_{b_n}\]
If I recall correctly, \(a_n\to\ln2\), so its partial sums must be bounded. \(b_n\) seems to be monotonic, and as \(n\to\infty\) you have \(b_n(x)\to1\) (whether this is uniform, I'm not sure).

Answer 3

No.. b_n(x) must tend to 0.@SithsAndGiggles

Answer 4

@SithsAndGiggles

Answer 5

@ganeshie8

Answer 6

@SithsAndGiggles I used Dirichlet's test too. however I took a_n=(-1)^n and b_n the rest./ Thanks for the idea

Answer 7

Yeah sorry, I misread the conditions in the link :/