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## Homework Statement

Show that the power series [itex]\sum_{k=1}^{k=\infty} \frac{x^{2k+1}}{k(2k+1)}[/itex] converges uniformly when [itex]|x| \leq 1[/itex]and determine the sum (at least when [itex]|x| < 1[/itex]).

## The Attempt at a Solution

Couldn't I somehow go about and show that, as [itex]|x| \leq 1[/itex], then [itex]f = \frac{x^{2k+1}}{k(2k+1)} \leq \frac{1^{2k+1}}{k(2k+1)} = \frac{1}{k(2k+1)} = g < \frac{1}{2k^2}[/itex] which converges. Then by Weierstrass majorant principle f should converge. Or am I missing something? And how do I find this sum?