Ramanujan sum of natural numbers – intuitive conundrum!

Srinivasa Ramanujan

Srinivasa Ramanujan FRS (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work

Here’s an amazing mathematical result derived by using Ramanujan summation:

1 + 2 + 3 + 4 + 5 + ... = -1/12 (R)


Intuitively, we keep thinking that this must be a really large number (mathematically called a divergent series), right? How can this result be negative number?

Here is a simple way to explain the proof without using any fancy mathematical jargon:

For more such fun with numbers, check Numberphile.com

Ramanujan wrote about this in his letter to G. H. Hardy, dated 27 February 1913:

Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich’s Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter…

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