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:
$$\sum_{k=0}^{\infty}\frac{(2k)!!}{(2k+1)!!}x^{2k+1}.$$

Try to find out $\lim {a_{n+1}\over a_n}$, $a_n={1.2.\dots 2n \over 1.3.5\dots 2n+1}x^{2n+1}$
