# 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}