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$\displaystyle f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$ would you tell me for what value of $x$ does the series converge uniformly? On what interval does it fail to converge uniformly and absolutely? Is $f$ continuous when the series converges? Is $f$ bounded?


I just able to show that when $x=-1/n^2$ It has problem. will be pleased for answer.

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    yes I know that M-test2012-04-26

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Taking one question more out of the unanswered questions' mud: $\forall\,0\neq x\in\mathbb R\,\,,\,\,1+n^2x>n^2x\Longrightarrow \frac{1}{1+n^2x}\leq\frac{1}{x}\frac{1}{n^2}$

Now use Weierstrass's M-test. Note that for $\,x=0\,$ the series trivially diverges.