# 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.