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There is a popular claim going around the internet that the sum of the positive integers is $-1/12$. There are proofs to this statement and I am not going to try and refute them. But I could not understand at what point the sum starts becoming more negative, since when I add positive numbers, the sum just gets more positive.

nathan.j.mcdougall
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    The sum you are referring to has a very technical way of interpreting that sum via an "analytic continuation". When it comes to infinite sums there are different conventions for interpreting them which give different results. Your intuition isn't wrong you're just using a different convention for how to interpret the sum. – Spencer Oct 27 '14 at 05:52
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    This isnt the "real" summation, it is a special way to value divergent sums. It is named Ramanujan summation. – Masacroso Oct 27 '14 at 05:52
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    Also, this is even similar to "When does $\sum\frac1{n^2}$ start getting irrational?" - It doesn't start getting so, it just is in the end – Hagen von Eitzen Oct 27 '14 at 06:39

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