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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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By the arithmetic-geometric mean inequality, $(1 + k x^2)/2 \ge |x| \sqrt{k}$, so $\left| \frac{x}{k(1+kx^2)}\right| \le \frac{2}{k^{3/2}}$.

This makes it easier, since $\sum k^{-3/2}$ converges.

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Thanks marty. Does this imply that the series converges uniformly on all of $\mathbb{C}$? My thinking is since it's absolutely convergent, then it's ordinarily convergent, so I would always be able to find partial sums large enough that their difference is smaller than some given $\epsilon>0$. – Dedede Jan 30 '12 at 5:31
That inequality fails when $\: k=1 \:$ and $\;\; x \: = \: \frac12 \cdot i \;\;$. $\;\;\;\;\;$ – Ricky Demer Jan 30 '12 at 8:59

That means determine the subsets of the complex plane on
which the sequence of partial sums is uniformly convergent.
For example, one conceivable answer would be "exactly the bounded subsets".
"uniform convergence at $x$" does not make sense, and
uniform convergence on $\{x\}$ is equivalent to convergence at $x$.

Perhaps, but the general rule is that if a sequence of continuous functions converges uniformly
on an open set, then the function defined by the limit is continuous on that open set.

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Thanks. So is there some systematic way to determine on which subsets the sequence of partial sums is uniformly convergent? – Dedede Jan 30 '12 at 5:13
Basically, find out an "essentially tight" modulus of convergence, where "essentially tight" means whatever you need it to mean to prove that uniform convergence fails on all other sets. $\;$ – Ricky Demer Jan 30 '12 at 9:01

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