Inspired by Patrick Corn

$\large 1-2+3-4+5-6+\cdots$

The series $\displaystyle \sum_{j=1}^{\infty} a_j$ is said to be Cesàro summable , with Cesaro Sum $A$ , if the average value of its partial sums $\displaystyle s_k=\sum_{j=1}^k a_j$ tends to $A$ , meaning that $A=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ns_k$

Is the series $1-2+3-4+5-6 + \cdots$

Cesàro summable? If so, enter its Cesàro sum $A$ . If not, enter 666.

Inspiration .