TITLE: What and where in the notebooks of Ramanujan is this series?
QUESTION [14 upvotes]: The wikipedia page on Ramanujan contains the following series:
$$
1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}
$$
$$\sum_{n=0}^\infty(-1)^n(4n+1) \left[ \frac{(2n-1)!!}{(2n)!!}\right]^3=\frac{2}{\pi}$$
Unfortunately, there is no reference as to where I can find this series in his notebooks (or in the 5-volume Springer set, for that matter), which is basically what I am after.
Also, though I do now study math, I am still a freshman, so I don't know enough about infinite series to be able to classify this series. Does it belong to a type of series? Is there a general class of series to which this series belongs?
Generally, I am very keen to know more about this series.

REPLY [3 votes]: I'm seriously late for this party but,  to address one of your questions, there are infinitely many formulas
$$\sum_{n=0}^\infty\frac{An+B}{C^n} \left[ \frac{(2n-1)!!}{(2n)!!}\right]^3=\frac{1}{\pi}$$
where $A,B,C$ are algebraic numbers. A few others are
$$\begin{aligned}\frac{2\sqrt{2}}{\pi}&=\sum_{n=0}^\infty(-1)^n\frac{6n+1}{2^{3n}} \left[ \frac{(2n-1)!!}{(2n)!!}\right]^3\\
\frac{4}{\pi}&=\sum_{n=0}^\infty\frac{6n+1}{2^{2n}} \left[ \frac{(2n-1)!!}{(2n)!!}\right]^3\\
\frac{16}{\pi}&=\sum_{n=0}^\infty\frac{42n+5}{2^{6n}} \left[ \frac{(2n-1)!!}{(2n)!!}\right]^3\end{aligned}$$
and so on. They belong to Ramanujan's fourth class of pi formulas. (I discussed the Wikipedia example in my blog Ramanujan Once A Day.)