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Could you tell me why the series $\displaystyle{\sum _k \frac{1}{1+k^2x^2}}$ doesn't converge uniformly on $(0,1]$?

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Apply Cauchy criterion and consider $x_k=1/k$ – Norbert Jan 9 '12 at 18:28
When you say "tell me why the series ... doesn't converge uniformly" do you mean "prove that the series ... doesn't converge uniformly" or something else? – Rasmus Jan 9 '12 at 18:55
The glib answer is that it doesn't converge uniformly on $(0,1]$ because it does not converge at all at $x=0$. – Adam Jan 9 '12 at 19:13
If $x$ is close to $0$, we have to go a ridiculously long way out to have the partial sum anywhere close to the full sm. – André Nicolas Jan 9 '12 at 21:48
@Norbert: You meant this: like in the answer we should have that $\frac{1}{1+N^2x^2}<\varepsilon$ (for a fixed $\varepsilon<\frac{1}{2}$), but if we take $x=\frac{1}{N}$ we obtain a contradiction. Is this you argument? – John Jan 9 '12 at 21:59
up vote 5 down vote accepted

Because, if $x\rightarrow 0$ for fixed $k$, the individual terms tend to $1$.

For uniform convergence you need to have that, independently of the choice of $x$, for each $\varepsilon >0$ there is $N\in \mathbb{N}$ such that $|\sum_{k=n}^{m} a_k(x) | < \varepsilon$ for $n,m \ge N $. Suppose such an $N$ exists. Now choose $\varepsilon = 1/2$ and look at $$ \sum_{N}^{N}a_k = \frac{1}{1+N^2 x^2}.$$ If $x\rightarrow 0$ this tends to $1 > \varepsilon$.

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