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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.

davidlowryduda
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ixtmixilix
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2 Answers2

11

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,

Raymond Manzoni
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3

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.)

Tito Piezas III
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