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I'm attempting to learn some complex analysis on my own, and I've run across a curious question.

My book says to "discuss" the uniform convergence of the series $\displaystyle\sum_{k=1}^\infty\frac{x}{k(1+kx^2)}$ for real values $x$.

I interpret this to determine the values of $x$ where the series is uniformly convergent, and I assume that means when the sequence of partial sums is uniformly convergent.

I define a sequence of functions $\{s_n(x)\}$ defined by $$ s_n(x)=\sum_{k=1}^n\frac{x}{k(1+kx^2)}. $$ Now for any $\epsilon>0$, I think I would like to find an $n_0$ such that for all $m\geq n\geq n_0$, $$ |s_m(x)-s_n(x)|=\left|\sum_{k=n+1}^m\frac{x}{k(1+kx^2)}\right|<\epsilon. $$

Am I correct in thinking that the $x$ which satisfy this for all $\epsilon$ will be the $x$ where the series is uniformly continuous? If so, how could I do this? I hope I have not interpreted the problem wrongly. Thanks kindly for your aid.

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