Fibonacci Sum

Relevant wiki: Sums Of Divergent Series

Let $S=1+1+2+3+5+8+\cdots$ , then
$\begin{aligned} 0&=S-S\\ &=(1+1+2+3+5+8+\cdots)-(1+1+2+3+5+8+\cdots)\\ &=1+(1+2+3+5+8+\cdots)-(1+1+2+3+5+8+\cdots)\quad\text{(stability)}\\ &=1+(0+1+1+2+3+\cdots)\quad\text{(linearity)}\\ &=1+S \end{aligned}$ So $S=-1$

Another method is to use Abel sum.
$\displaystyle S=\lim_{x\to 1^-}(1+x+2x^2+3x^3+\cdots)=\lim_{x\to 1^-}\frac{1}{1-x-x^2}=-1$