Is $2 + 2 + 2 + 2 + ... = -\frac12$ or $-1$?

divergent-series regularization

Solution 1:

ans: $-1$

Any linear summation method should have $\displaystyle \sum_{n=1}^{\infty}a_nk=k\sum_{n=1}^{\infty}a_n$.

A regular linear stable summation method need not give the same answer when $2$ is rewritten as $1+1$ for infinitely many of the $2$s, considering that infinitely permuting terms in diverging series can alter the sum, and that in $1+2+4+8+...=-1$, setting $2=1+1, 4=1+1+1+1$ etc. would make the sum $-\frac{1}{2}$.

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