Strong Cesàro summability

Remarks

Example: a convergent series with sum $s$ is strongly Cesàro summable to $s$ for every $q > 0$. Indeed $\abs{s_k - s}^q \to 0$, and the arithmetic means of a sequence tending to $0$ also tend to $0$.

Non-example: Grandi's series $1 - 1 + 1 - \cdots$ is Cesàro summable to $1/2$, but it is not strongly Cesàro summable with any exponent. Its partial sums alternate between $1$ and $0$. For any $s$, one of $\abs{1 - s}$ and $\abs{s}$ is at least $1/2$. So at least half of the terms $\abs{s_k - s}^q$ are at least $2^{-q}$, and the means do not tend to $0$.

For $q \ge 1$, the power mean inequality gives $$\abs{\frac{1}{n+1} \sum_{k=0}^{n} (s_k - s)} \le \left( \frac{1}{n+1} \sum_{k=0}^{n} \abs{s_k - s}^q \right)^{1/q},$$ so strong Cesàro summability to $s$ implies Cesàro summability to $s$.