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Wiki states that:

The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a L'Hôpital's rule for sequences.

To be precise, Stolz–Cesàro states that:

If $$\lim_\limits{n\to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1}- b_{n}} = \lim\limits_{n\to \infty} \frac{a_{n}}{b_{n}}=k.$$

In contrast L'Hôpital's Rule states that:

$$ \lim_\limits{x\to \infty} \frac{f^{(n)}(x)}{g^{(n)}(x)}= \lim\limits_{x\to \infty} \frac{f(x)}{g(x)}= k.$$

Although some similarity is apparent, I can't see how one generalizes the other. L'Hôpital's rule is about derivatives of the numerator and denominator when Stolz-Cesaro is about sub-sequences.

amWhy
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HelloWorld
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2 Answers2

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Define the forward difference $\Delta_h[f](x)=f(x+h)-f(x)$. Dropping the square brackets, the derivative of $f$ is $Df:=\lim_{h\to0}\frac{\Delta_hf}{h}$. Stolz–Cesàro says $\frac{\frac{\Delta_ha}{h}}{\frac{\Delta_hb}{h}}\sim\frac{a}{b}$ for $h=1$, where $\sim$ denotes matching $n\to\infty$ limits. L'Hôpital says $\frac{\lim_{h\to0}\frac{\Delta_ha}{h}}{\lim_{h\to0}\frac{\Delta_hb}{h}}\sim\frac{a}{b}$ (if you'll let me rename $f$ as $a$ & $g$ to $b$ for a more obvious comparison), where $\sim$ now denotes matching $x\to\infty$ limits. (This is only the $n=1$ case, but you can imply equivalence $n$ times with either theorem.)

J.G.
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The discrete analogue of the $n^{th}$ order derivative is the $n^{th}$ order finite difference. The generalization is obvious.