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The infinite sum of the positive integers

The infinite sum of the positive integers.

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The infinite sum of the positive integers

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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. … – SrinivasaRamanujan, writing to G. H. Hardy, Feb 1913  

  2. Know your shapes in topology! The same letter indicates the same point. Parallel arrows that connect the same two points represent the same edge. A B 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 A A A A A A A A Möbius Strip Sphere A B A B B A C A

  4. Tube A A B B

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