# Inspired by Patrick Corn, again

Calculus Level 5

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

For a series $$\displaystyle \sum_{n=1}^{\infty} a_n$$ we define its partial sums $$\displaystyle s_n=\sum_{k=1}^n a_k$$, their average values $$\displaystyle b_n=\frac{1}{n}\sum_{k=1}^n s_k$$, and the average values of those, $$\displaystyle c_n=\frac{1}{n}\sum_{k=1}^n b_k$$.

Let's say that the series $$\displaystyle \sum_{n=1}^{\infty} a_n$$ is Corn summable if $$\displaystyle \lim_{n\to\infty}c_n$$ exists. In this case, $$\displaystyle\lim_{n\to\infty}c_n$$ is called the Corn sum of the series.

Find the Corn sum of the series $1-2+3-4+ 5 - 6+\cdots$

If you come to the conclusion that no such sum exists, enter 0.666.

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