Question

prove that 1+2+3+4+5...=-1/12

Answer 1

I have seen this once before with the Riemann Zeta function, for what do you need it?

Answer 2

this is not true for the standard meaning of \(=\) in fact it's meaningless in that sense since \(1+2+3+\dots\) is clearly divergent

Answer 3

it is true that \(\zeta(-1)=-\frac1{12}\) but note for \(\Re\{s\}\le0\) this surely is not the definition you are invoking:$$\zeta(s)=\sum_{n=1}^\infty n^{-s}$$instead, it's an *analytic continuation* of this function to cover a broader domain