Ramanujan's Hypergeometric Identity

$$1-\left({1\over 2}\right)^3+\left({1\cdot 3\over 2\cdot 4}\right)^3+\ldots = {}_3F_2\left({\begin{array}{c}{\textstyle{1\over 2}}, {\textstyle{1\over 2}}, {\textstyle{1\over 2}}\\ 1, 1\end{array} ; -1}\right)$$

$$= \left[{{}_2F_1\left({\begin{array}{c}{\textstyle{1\over 4}}, {\textstyle{1\over 4}}\\ 1\end{array}; -1}\right)}\right]^2$$

$$= {\Gamma^2({\textstyle{9\over 8}})\over \Gamma^2({\textstyle{5\over 4}})\Gamma^2({\textstyle{7\over 8}})},$$

where ${}_2F_1(a,b;c;x)$ is a Hypergeometric Function, ${}_3F_2(a,b,c;d;e;x)$ is a Generalized Hypergeometric Function, and $\Gamma(z)$ is a Gamma Function.

References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 106, 1959.