Evaluate this power series Evaluate the sum
$$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$
Totally no idea. I think this series may related to the $\sin x$ series because of those missing even powers. Another way of writing this series:
$$\sum_{k=0}^{\infty}\frac{(2k)!!}{(2k+1)!!}x^{2k+1}.$$

I refer you to this solution.  The function $f(x)$ represented by the sum in question is
$$f(x) = \sum_{n=0}^{\infty} \frac{2^{2 n}}{\displaystyle (2 n+1) \binom{2 n}{n}} x^{2 n+1} = \frac{\arcsin{x}}{\sqrt{1-x^2}}$$
You may deduce that the radius of convergence is $1$ from the relation
$$\frac{1}{2^{2 n}} \binom{2 n}{n} \sim \frac{1}{\sqrt{\pi n}} \quad (n \to \infty)$$
which may be shown as a result of, e.g., Stirling's approximation.
