4 Jul
2006
4 Jul
'06
4:33 p.m.
Marc writers: << This summer's AMS Notices reproduces Ramanujan's letter to Hardy where he asserts 1 + 2 + 3 + 4 + ... = - 1/12 On the hypothesis that even the errors of genius may hold value, I've always been curious what his theory was under which this result was derived.
I've seen it written in several places that since 1+2+3+ . . . = sum{n=1,oo} of 1/n^(-1), Ramanujan thought of this sum, in some sense, as zeta(-1), which turns out (using correct analytic continuation of the Dirichlet series zeta(s) = sum{n=1,oo} of 1/n^s, which converges only for Re(s) > 1) to be -1/12. --Dan