Stolz-Cesaro Theorem

There are several version of this theorem. We start with one that shows the intuition behind it.

Theorem (Stolz-Cesaro).
Let {a_n}_n≥n0 and {b_n}_n≥n0 be sequences of positive numbers. Assume that  ∑ b_n = ∞. If the sequence {a_n / b_n} has a limit, then

Why should this be true? Let's explore the case when the ratio a_n / b_n has a finite limit L. Then for large n we have a_n / b_n ∼ L. This means that a_n ∼ L⋅b_n. Consequently,

One problem si that we can replace a_i with L⋅b_i only when i is large. This is where the other assumption comes in. The fact that sum of b_i is infinite means that first several terms in the two sums do not play any role in the final outcome.

The above theorem can be stated in a different way. This brings us to its traditional form.

Theorem (the Stolz-Cesaro theorem).
Let {a_n} and {b_n} be sequences of real numbers. Assume that {b_n} is strictly increasing and goes to infinity. If the limit

exists, then

This theorem also applies to cases when {b_n} is strictly decreasing and goes to negative infinity, as we can see by considering −a_n and −b_n. How do we know that the two theorems above in fact say the same thing? Denote A_n = a_n+1 − a_n and B_n = b_n+1 − b_n. The validity of the Stolz-Cesaro theorem then follows by applying the first theorem above to numbers A_n, B_n and vice versa.

One can see this theorem as a twin of l'Hospital rule in the world of sequences. Indeed, imagine that the sequences are given by functions as in section Sequences and functions. In particular, assume that a_n = f (n). We could pass to limit of function and l'Hospital's rule would require knowledge of derivative f ′. This derivative can be approximated with ratios

We expect to get better approximation when we take smaller h. If we look at this situation through the eyes of sequences, we are allowed to substitute only integers, and the smallest h is therefore 1:

Thus the ratio on the right in the above theorem is an approximate replacement for ratio of derivatives in l'Hospital's rule. Of course, there are some rather large holes in this argument, so this is just a story that sheds some light on the situation, not a proper mathematical treatment. In any case, we follow the similarity trail by stating also the "zero over zero" version.

Theorem.
Let {a_n} and {b_n} be sequences of real numbers. Assume that {b_n} is strictly monotone and both sequences go to zero. If the limit

exists, then

Unlike l'Hospital's rule, the Stolz-Cesaro theorem does not help much when solving everyday limit problems. It can be very useful when dealing with certain special limits and above all, it has some interesting theoretical consequences and some useful versions and corollaries.

Corollary.
Let {a_n} be a sequence of real numbers. If it has a limit, then

Corollary.
Let {a_n} be a sequence of positive real numbers. If it has a limit, then

Corollary.
Let {a_n} be a sequence of positive real numbers. If {a_n+1 / a_n} has a limit, then

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