God Plays Dice: The Stolz-Cesàro theorem

09 May 2008

The Stolz-Cesàro theorem

The Stolz-Cesàro theorem is a discrete analogue of L'Hôpital's rule, with which I was not familiar.

(From Topological Musings.)

4 comments:

Daniel Andrés said...: I didn't understand the last equality before taking the limit in the first problem :-|. Anyway, it is a nice result.; May 9, 2008 at 12:31 PM
Anonymous said...: Daniel: Allow me to explain that part. In the first problem, when evaluating the limit, we notice that the denominator is a polynomial in n. (k is just a constant.) And since the term n^k in the denominator dominates the other polynomial terms, it makes sense to divide both the denominator and the numerator by n^k to get a constant term (k+1, k) in the denominator. (k+1, k) is just "k+1 choose k" or number of ways of choosing k objects out of k+1 distinct objects. And then in the last step, when taking limits, the numerator just goes to 1 while the denominator stays k+1.

Hope this helps. Thanks!; May 9, 2008 at 2:52 PM
misha said...: To prove Stolz-Cesàro, you can extend the sequences to piecewise-linear functions by linearly interpolatiing between n and n+1, and then notice that L’Hospital’s rule holds for continuous piecewise-differentiable functions.; May 9, 2008 at 6:52 PM
Daniel Andrés said...: Thanks for the explanation, Vishal!; May 10, 2008 at 8:04 AM

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