<P> Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = − 1 / 12" in chapter 8 of his first notebook . The simpler, less rigorous derivation proceeds in two steps, as follows . </P> <P> The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 1 − 2 + 3 − 4 + ⋯ . The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century . </P> <P> In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on . The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series . These relationships can be expressed using algebra . Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯ . Then multiply this equation by 4 and subtract the second equation from the first: </P> <Dl> <Dd> c = 1 + 2 + 3 + 4 + 5 + 6 + ⋯ 4 c = 4 + 8 + 12 + ⋯ c − 4 c = 1 − 2 + 3 − 4 + 5 − 6 + ⋯ (\ displaystyle (\ begin (alignedat) (7) c& () = () &1 + 2&& () + 3 + 4&& () + 5 + 6 + \ cdots \ \ 4c& () = () &4&& () + 8&& () + 12 + \ cdots \ \ c - 4c& () = () &1 - 2&& () + 3 - 4&& () + 5 - 6 + \ cdots \ \ \ end (alignedat))) </Dd> </Dl>

Why the sum of all positive integers is 1 12