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According to Wikipedia, given a series $\sum a_n$ we can define a general Cesàro sum (C, $\alpha$) for $\alpha \in \Bbb R \setminus \Bbb N$ as $\lim_{n\to\infty}\dfrac {A^\alpha_n}{E^\alpha_n}$ where

$$\sum_{n=0}^{\infty} A^\alpha_nx^n = \frac{\sum_{n=0}^{\infty} a_nx^n}{(1-x)^{1+\alpha}}$$

and $E_n^\alpha$ are the binomial coefficients of power $-1-\alpha$.

I am interested in seeing why for $ \alpha <\beta$, the (C, $\beta$) sum is stronger than the (C, $\alpha$) sum, possibly with examples.

In a particular case, when $\alpha$ is a positive integer, they claim that this is equivalent to the $\alpha$ iteration of the usual Cesàro sum. Assuming that this is true, it's easy to see that iterating the usual Cesàro sum can only yield a stronger summation method. Maybe there are easier examples here of how the summation becomes strictly stronger, as the Grandi's series is an example of a series that doesn't converge, but is Cesàro (C, $1$) summable.

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