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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

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Know your shapes in topology! AB 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).

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Klein Bottle Torus Sphere M ö bius Strip A A AA AA AA AB AC AB AB

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Tube AA BB

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Section 3.3: Mathematical Induction Mathematical induction is a proof technique that can be used to prove theorems of the form: n Z +,P(n) We have.

Section 3.3: Mathematical Induction Mathematical induction is a proof technique that can be used to prove theorems of the form: n Z +,P(n) We have.

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