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I have a question in baby rudin p.165, exercise #4. The problem is :

Consider f(x)=$\sum_{n=1}^\infty\frac{1}{1+n^2x}$

For what values of x does the series converge absolutely? On what intervals does it converge uniformly? On what intervals does it fail to converge uniformly? Is f continuous wherever the series converges? Is f bounded?

I don't know how to solve this problem. If you solve it, I appreciate you very much. Thank you ! :)

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Try comparison test with $\frac{1}{x}\sum_{n=1}^{\infty}\frac{1}{n^2}$ for $x\neq 0$, at $x=0$ judge convergence by inspection.

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