In mathematics, the StolzâÂÂCesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.
The StolzâÂÂCesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
Let and be two sequences of real numbers. Assume that is a strictly monotonic and divergent sequence (i.e. strictly increasing and approaching , or strictly decreasing and approaching ) and the following limit exists:
Then, the limit
Let and be two sequences of real numbers. Assume now that and while is strictly decreasing. If
then
Case 1: suppose strictly increasing and divergent to , and . By hypothesis, we have that for all there exists such that
which is to say
Since is strictly increasing, , and the following holds
Next we notice that
thus, by applying the above inequality to each of the terms in the square brackets, we obtain
Now, since as , there is an such that for all , and we can divide the two inequalities by for all
The two sequences (which are only defined for as there could be an such that )
are infinitesimal since and the numerator is a constant number, hence for all there exists , such that
therefore
which concludes the proof. The case with strictly decreasing and divergent to , and is similar.
Case 2: we assume strictly increasing and divergent to , and . Proceeding as before, for all there exists such that for all
Again, by applying the above inequality to each of the terms inside the square brackets we obtain
and
The sequence defined by
is infinitesimal, thus
combining this inequality with the previous one we conclude
The proofs of the other cases with strictly increasing or decreasing and approaching or respectively and all proceed in this same way.
Case 1: we first consider the case with and strictly decreasing. This time, for each , we can write
and for any such that for all we have
The two sequences
are infinitesimal since by hypothesis as , thus for all there are such that
thus, choosing appropriately (which is to say, taking the limit with respect to ) we obtain
which concludes the proof.
Case 2: we assume and strictly decreasing. For all there exists such that for all
Therefore, for each
The sequence
converges to (keeping fixed). Hence
and, choosing conveniently, we conclude the proof
The theorem concerning the case has a few notable consequences which are useful in the computation of limits.
Let be a sequence of real numbers which converges to , define
then is strictly increasing and diverges to . We compute
therefore
<blockquote>Given any sequence of real numbers, suppose that
exists (finite or infinite), then
Let be a sequence of positive real numbers converging to and define
again we compute
where we used the fact that the logarithm is continuous. Thus
since the logarithm is both continuous and injective we can conclude that
<blockquote>Given any sequence of (strictly) positive real numbers, suppose that
exists (finite or infinite), then
Suppose we are given a sequence and we are asked to compute
defining and we obtain
if we apply the property above
This last form is usually the most useful to compute limits <blockquote>Given any sequence of (strictly) positive real numbers, suppose that
exists (finite or infinite), then
where we used the representation of as the limit of a sequence.
The âÂÂ/â case is stated and proved on pages 173âÂÂ175 of Stolz's 1885 book and also on page 54 of Cesàro's 1888 article.
It appears as Problem 70 in Pólya and Szegà  (1925).
The general form of the StolzâÂÂCesàro theorem is the following: If and are two sequences such that is monotone and unbounded, then:
Instead of proving the previous statement, we shall prove a slightly different one; first we introduce a notation: let be any sequence, its partial sum will be denoted by . The equivalent statement we shall prove is: <blockquote>Let be any two sequences of real numbers such that
then
</blockquote>
First we notice that:
Therefore we need only to show that . If there is nothing to prove, hence we can assume (it can be either finite or ). By definition of , for all there is a natural number such that
We can use this inequality so as to write
Because , we also have and we can divide by to get
Since as , the sequence
and we obtain
By definition of least upper bound, this precisely means that
and we are done.
Now, take as in the statement of the general form of the Stolz-CesÃÂ ro theorem and define
since is strictly monotone (we can assume strictly increasing for example), for all and since also , thus we can apply the theorem we have just proved to (and their partial sums )
which is exactly what we wanted to prove.