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, by the Stolz- Cesàro theorem, we have
, and so