# A minimalist proof of $1+2+3+...=-1/12$ \begin{aligned} S_1 &= 1-1+1-1+\ldots \\ 1-S_1 &= 1-(1-1+1-1+\ldots) \\ &= 1-1+1-1+\ldots = S_1\\ S_1 &=\frac{1}{2} \end{aligned} \begin{aligned} S_2 &= 1-2+3-4+\ldots \\ 2S_2 &= 1-2+3-4+\ldots \\ &+\quad\quad 1-2+3-4+\ldots \\ &= 1-1+1-1+\ldots = S_1 \\ S_2 &=\frac{1}{4} \end{aligned} \begin{aligned} S &= 1+2+3+4+\ldots \\ S-S_2 &= 1+2+3+4+\ldots \\ &- (1-2+3-4+\ldots) \\ &= 4(1+2+3+4\ldots) = 4S \\ S &=-\frac{1}{3}S_2=-\frac{1}{12} \end{aligned}