Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ How does one sum the series $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$
This was asked to me by a high school student, and I am embarrassed that I couldn't solve it. Can anyone give me a hint?!

Making my comments more explicit:
Your sum of interest is
$\sum_{j=0}^\infty {(-1)^j \frac{(2j)!!}{(2j+1)!!} a^{2j+1}}$
where $(2j)!!=2\cdot 4\cdot 6\cdots (2j)$ and $(2j+1)!!=3\cdot 5\cdot 7\cdots (2j+1)$.
To simplify things a bit, we rearrange the series a bit to
$a\sum_{j=0}^\infty {\frac{(2j)!!}{(2j+1)!!}\left(-a^2\right)^j}$
The double factorials can be also expressed as
$(2j)!!=2^j j!=2^j (1)_j$
and
$(2j+1)!!=2^j \left(\frac32\right)_j$
where $(a)_j$ is a Pochhammer symbol.
Substitute both expressions into the series, and then note that the series now looks like a hypergeometric series. Now you can employ the formula here.
