I am giving a lecture to undergraduates on the lovely identity $$1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}.$$
Ramanujan wrote in his second letter to Hardy (courtesy Wikipedia),
"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. …"
Does anyone know where I might find a scanned copy of this or closely related material, in Ramanujan's handwriting? Bruce Berndt kindly pointed me to a typeset version, and much else related of extreme interest — but I would love to use the original in my lecture, if possible.
Thank you!
Best Answer
This article, by Stephen Wolfram from his book Idea Makers, has 9 photo images of Ramanujan's letter to Hardy. Not quite sure if it is the whole of his second letter, but it has the famous -1/12 formula on it.