$\pi$ as a product of a fraction and $\sqrt2$

Question

I have read that Ramanujan expressed $\pi$ as a product of the square root of 2 and a fraction, but I cannot find the book now. Is this true? If it is, then what is the fraction? Can you please show me a proof if not? Thanks in advance.

Answer 1

$$\begin{align} \frac{9801}{4412}\sqrt{2} &= 3.14159273... \\ \pi &= 3.14159265... \end{align} $$

This comes from the first term of Ramanujan series $$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^\infty\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}.$$ By taking more terms we can get better approximations. For example, with two terms we get $$\begin{align} \frac{2510613731736}{1130173253125}\sqrt{2} &= 3.1415926535897939... \\ \pi &= 3.1415926535897932... \end{align} $$ But we will never get exactly $\pi$, since $\pi$ is a transcendental number.