Take away a third at a time

Calculus Level 3

\[\large 1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + \cdots \]

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 \(\displaystyle A=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ns_k\).

Is the series above Cesàro summable?

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