“How he came up with those non-obvious numbers?”. No-one knows. We just know that he was far ahead from other mathematicians in understanding of elliptical functions, theta function and modular equations in an intuitive fashion. No one knows how he reached to the numbers like 26390, 1103 etc. Modern approach uses numerical calculations for the same. Here you can find some relationship of the above mentioned numbers with Pell Equations.
“Does the equation has any relation with circle or trigonometric function. If it is not related, then from where π came from in this formula”. At first sight, the series might not look to have any relationship with π any trigonometric functions but that’s not true. All trigonometric, exponential, hyperbolic, elliptical etc. functions can be written in form of infinite series (Taylor Expansions). This series in one of many approximation series that he gave to calculate π. His theory for was that we can have infinitely many series of form.
1/π=∑(a+bn)c^n n=(0 to infinity)
and the form mentioned in the question is one of these.
“Does it have any proof”. I was not able to find any text for a valid proof, but that is limitation of my search as I have found mention of its proofs in some online articles.
However, the man was an amazing intuitive mathematician, autodidact, even there is a legend component because Ramanujan said that the goddess Namagiri told him the solutions in dreams:
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