Run out of sums

Calculus Level 3

77+7374+75\large 7 -\sqrt7 + \sqrt[3]{7} - \sqrt[4]{7} + \sqrt[5]{7} - \cdots

The series j=1aj\displaystyle \sum_{j=1}^{\infty} a_j is said to be Cesàro summable, with Cesaro Sum AA, if the average value of its partial sums sk=j=1kaj\displaystyle s_k=\sum_{j=1}^k a_j tends to AA, meaning that A=limn1nk=1nsk\displaystyle A=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ns_k .

Is the series above Cesàro summable?

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