<P> The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also − 1 / 12 . Ramanujan wrote in his second letter to G.H. Hardy, dated 27 February 1913: </P> <Dl> <Dd> "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. ..." </Dd> </Dl> <Dd> "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. ..." </Dd> <P> Ramanujan summation is a method to isolate the constant term in the Euler--Maclaurin formula for the partial sums of a series . For a function f, the classical Ramanujan sum of the series ∑ k = 1 ∞ f (k) (\ displaystyle \ sum _ (k = 1) ^ (\ infty) f (k)) is defined as </P>

Sum of all positive integers from 1 to infinity