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# System prepended metadata

title: A minimalist proof of $1+2+3+...=-1/12$

---

# 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}