I wanted to know how the Ramanujan series works only using basic calculus. Why is it a shocking fact for the sum of an infinite series to be $-\frac1{12}$? How is it important to us and how does it change our perspective towards mathematics? Have his theories ever been proved wrong? I'm sorry for ranting, but this topic intrigues me and I couldn't find any satisfactory answers on google.
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You could take a look at Mathologer's video on YouTube... – Riccardo Orlando Apr 30 '16 at 06:22
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First of all, the sum is $- \frac{1}{12}$. This is more than a little surprising because 1) all the terms of the sum are integers, 2) they are all positive integers, even though the answer is negative, and 3) the partial sums are not only all positive integers but they are increasing positive integers, tending to infinity... – Qiaochu Yuan Apr 30 '16 at 06:25
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Wait, but doesn't $1+2+3+4\ldots$ not converge? – Frank Nov 01 '16 at 00:29