What and where in the notebooks of Ramanujan is this series?
As suggested by Anon it appears in the "Hypergeometric series" chapter of Hardy's book "Ramanujan" (formula (1.2)-(1.4) of page 7 and (7.4.2) and others of page 105). You may 'look inside' Hardy's book at amazon.
Ramanujan published an article in 1914 "Modular Equations and Approximations to $\pi$" that contains some related examples (28),(29),...
The formula appears in Berndt's "Ramanujan's Notebooks II" page 23s (search 'Example' and click on 'page 16').
A search of Dougall-Ramanujan Identity and variants could be helpful too since your equation is a special case of this identity (follow the Morley's formula link for another simple example).
See the Tract of Bailey 'Generalized Hypergeometric Series' for many more hypergeometric identities and the examples provided page 96.
Hoping it helped,
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.)