The Infinite Sum of Positive Integers
A letter expressing gratitude for a correspondence about the mathematical concept of adding all positive integers, received on a certain date.
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1. The infinite sum of the positive integers 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. Srinivasa Ramanujan, writing to G. H. Hardy, Feb 1913
2. Know your shapes in topology ! A B The same letter indicates the same point. Parallel arrows that connect the same two points represent the same edge. A dotted line indicates edges which are free (i.e. We can keep going in that direction without reaching another edge).
3. Klein Bottle Torus Sphere M bius Strip A A A A A A A A A B A C A B A B
4. Tube A A B B