In his paper Modular equations and approximations to $\pi$ Ramanujan gave the following spectacular series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} +
\frac{27493}{99^{6}}\frac{1}{2}\frac{1\cdot 3}{4^{2}} +
\frac{53883}{99^{10}}\frac{1\cdot 3}{2\cdot 4}\frac{1\cdot 3\cdot 5\cdot
7}{4^{2}\cdot 8^{2}} + \cdots\tag{1}$$ without any proof. The series converges very fast with each term giving almost 8 digits of the final sum. It was proved by Borwein brothers in 1987 with some calculations assisted via computer software. And most recently in 2025 it was proved by Jesus Guillera using MAPLE software. There is no known proof using hand calculation available so far. Using ideas from a paper by Pablo F. Refolio, an independent researcher from Spain, I will present a proof which involves calculations via hand, pen and paper.
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