The following theorem, I feel, is not very well-known, though it is a particularly useful one for solving certain types of “limit” problems. Let me pose a couple of elementary problems and offer their solutions. First, the theorem.

**Stolz-Cesàro**: Let and be two sequences of real numbers, such that is positive, strictly increasing and unbounded. Then,

,

if the limit on the right hand side exists.

The proof involves the usual method, and I will avoid presenting it here since it isn’t particularly interesting. Just as *Abel’s lemma* is the discrete analogue of *integration by parts*, the *Stolz-Cesàro theorem* may be considered the discrete analogue of *L’Hospital’s rule* in calculus.

**Problem 1**: Evaluate the limit , where .

**Solution**: One may certainly consider the above limit as a Riemann-sum which may then be transformed into the integral , which then obviously evaluates to . But, we will take a different route here.

First, let and . Then, we note that the sequence is positive, strictly increasing and unbounded. Now,

(using the binomial theorem)

.

Therefore, using the Stolz-Cesàro theorem, we conclude that the required limit is also .

Let us now look at another problem where applying the aforesaid theorem makes our job a lot easier. This problem is an example of one that is not amenable to the other usual methods of evaluating limits.

**Problem 2**: Let be integers and suppose . Given the tangent line at the point from the point to , evaluate

.

**Solution**:(This is basically the solution I had offered elsewhere a while ago; so, it’s pretty much copy/paste!)

.

So, the equation of the tangent line at the point is given by

Since the point lies on this line, we must have

The above, after squaring and some algebraic manipulation yields

, which implies . We drop the negative root because for all .

(This is where the Stolz-Cesàro theorem actually comes into play!)

Now, let and be two sequences such that and

Note that is a positive, increasing and unbounded sequence.

Therefore,

.

Therefore, by the Stolz- Cesàro theorem, we have

, and so

.

## 19 comments

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May 8, 2008 at 5:15 pm

John Armstrongcite

September 19, 2012 at 10:41 am

Luqing YeAbel’s lemma can be used to prove integration by parts.Stolz’s theorem can be used to prove L’Hospital’s law.

May 8, 2008 at 11:40 pm

Todd TrimbleRe the proof of Stolz-Cesàro: there’s the fun little fact that if and , then

< <

(“the quotient sum of two fractions is between the two fractions”). It follows that if the fractions are in the interval for , then so are all the “telescoping quotient sums” . Since the grow without bound, is also in this interval for sufficiently large. The conclusion follows.

May 9, 2008 at 1:47 am

Vishal LamaTodd,

That is a wonderful proof! Thanks for sharing it. I hope you won’t mind if I incorporate the proof in the post some time later. I may perhaps expand a few steps just so that some of our readers may find it easier to see what you just did.

May 9, 2008 at 7:40 pm

John ArmstrongAnother consequence of the lemma Todd states is that batting averages actually behave like averages, in that they lie in the middle of the data.

May 10, 2008 at 1:14 am

mishaThere is still another neat way to figure out the limit in problem 1. It is based on the combinatorial identity that says that to pick k+1 numbers from 1,2,3,…,n you need first to pick your biggest number and then the rest k of them.

May 10, 2008 at 1:43 am

mishaStill another proof of Stolz-Cesàro. Extend the sequences to piecewise-linear functions by linearly interpolatiing between n and n+1. Then you can either notice that L’Hospital’s rule holds for comtinuous piecewise-differentiable functions, or smooth out the corners and use L’Hospital directly.

May 10, 2008 at 5:02 am

Vishal LamaMisha,

Those are awesome proofs! Thanks very much!

May 10, 2008 at 7:03 am

mishaStill another way to look at Stolz-Cesàro, that makes it sort of obvious. Begin by linear interpolation, like in my previous comment, then take b as the independent variable.

May 11, 2008 at 6:47 am

mishaThat made me think about the alternative proof of the L’Hospitals’s rule itself. Just take the denominator as the independent variable etc.

May 16, 2008 at 4:35 am

Ars Mathematica » Blog Archive » Stolz-Cesaro Theorem[…] Musings has a post up about a cute little result, the Stolz-Cesaro theorem. The result is a discrete analogue of l’Hôpital’s […]

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