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I want to determine if $$f(x)=\sum_{k=1}^\infty \frac{k^{2}x}{1+k^{4}x^{2}}$$ is continuous at x=0. I tried to use the definition: $\lim_{x \to 0} f(x)=f(0)$. Right side is equal to 0, but not sure how to evaluate $\lim_{x\to 0} \sum_{k=1}^\infty \frac{k^{2}x}{1+k^{4}x^{2}}$

Could anyone help me?

Is there maybe an easier way to show that it is continuous than my method?

José Carlos Santos
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fejz1234
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  • you have to argue why you can switch the limits $$\lim_{x\to 0} \lim_{n\to\infty} \sum_{k=1}^n \frac{k^2x}{1+k^4x^2} = \lim_{n\to\infty} \sum_{k=1}^n \lim_{x\to 0}\frac{k^2x}{1+k^4x^2} $$ therefore one limit has to converge uniformly – Nathanael Skrepek Mar 11 '17 at 16:29
  • Actually, I doubt if the claim is true. For $x$ small, there will be $k$ such that $k^2x$ will be between, say $1$ and $5$, this way, one summand will be a significant number, bigger than $1/26$. Making $x$ closer to $0$ will not help, as this will again happen, probably at a different $k$. – Behnam Esmayli Mar 11 '17 at 16:37
  • @NathanaelSkrepek Im not sure I understood your method, and what you meant with "therfore one limit has to converge uniformly" ? – fejz1234 Mar 11 '17 at 16:41
  • it is not always valid to switch limits! However under certain circumstances you can do without changing the result. One way to make sure that you are allowed to switch limits is to show that one limit converge uniformly and the other at least pointwise. – Nathanael Skrepek Mar 11 '17 at 16:53
  • http://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits there is a thread about it – Nathanael Skrepek Mar 11 '17 at 16:53

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Claim: The function is not continuous!

To see this, I claim that no matter what $\delta >0$ you take, there is always an $x$ such that $$ |x-0|<\delta,$$ but $$|f(x) - 0|>0.5$$ Why is this true? Simply take $x=1/k^2$ for large enough $k$ such that $1/k^2 < \delta$. Then the $k$-th summand is 0.5, and all others are positive.

This proves that the limit of $f$ cannot be zero at zero.

Behnam Esmayli
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  • Ok is this the standard-method to use when you have to determine if a sum is continuous at a specific point? Or are there more way to also solve it? – fejz1234 Mar 11 '17 at 17:38
  • To prove continuity of sums, I will take the direct path. I will try to put a bound on the error I commit in each summand by moving from x to a nearby x'. But I try to have these bounds so that their sum will converge to some bound that itself approaches zero as x' approaches x. – Behnam Esmayli Mar 11 '17 at 21:08