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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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    I had a nonsense answer which the user Henry fixed to the correct answer - hopefully he will come back and post it as an answer.2012-04-25
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    Extending Gingerjin's (and Henry's) hint: If $x>1/K>0$, then $0<(1+n^2x)^{-1}.2012-04-25
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    ... and criticizing the phrasing of the question a bit. A series does not converge uniformly at a single point. It simply converges (possibly absolutely) or diverges. Uniform convergence takes place (or not) on a set (typically an interval, but could be a more general set also).2012-04-25
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    Have you covered something called "Weierstrass' M-test" in class?2012-04-26
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    yes I know that M-test2012-04-26

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