Stolz–Cesàro theorem

In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.

The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

Statement of the theorem for the */∞ case

Let and be two sequences of real numbers. Assume that is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching , or strictly decreasing and approaching ) and the following limit exists:

Then, the limit

Statement of the theorem for the 0/0 case

Let and be two sequences of real numbers. Assume now that and while is strictly decreasing. If

then

[1]

Proofs

Proof of the theorem for the */∞ case

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.

Proof of the theorem for the 0/0 case

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

such that

and, choosing conveniently, we conclude the proof

Applications and examples

The theorem concerning the ∞/∞ case has a few notable consequences which are useful in the computation of limits.

Arithmetic mean

Let be a sequence of real numbers which converges to , define

then is strictly increasing and diverges to . We compute

therefore

Given any sequence of real numbers, suppose that

exists (finite or infinite), then

Geometric mean

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

.

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

Given any sequence of (strictly) positive real numbers, suppose that

exists (finite or infinite), then

Example 1

Example 2

where we used the representation of as the limit of a sequence.

History

The ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book and also on page 54 of Cesàro's 1888 article.

It appears as Problem 70 in Pólya and Szegő (1925).

The general form

Statement

The general form of the Stolz–Cesàro theorem is the following:[2] If and are two sequences such that is monotone and unbounded, then:

Proof

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:

Let be any two sequences of real numbers such that

  • ,
  • ,

then

Proof of the equivalent statement

First we notice that:

  • holds by definition of limit superior and limit inferior;
  • holds if and only if because for any sequence .

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.

Proof of the original statement

Now, take as in the statement of the general form of the Stolz-Cesàro 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.

References

  • Mureşan, Marian (2008), A Concrete Approach to Classical Analysis, Berlin: Springer, pp. 85–88, ISBN 978-0-387-78932-3.
  • Stolz, Otto (1885), Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten, Leipzig: Teubners, pp. 173–175.
  • Cesàro, Ernesto (1888), "Sur la convergence des séries", Nouvelles annales de mathématiques, Series 3, 7: 49–59.
  • Pólya, George; Szegő, Gábor (1925), Aufgaben und Lehrsätze aus der Analysis, vol. I, Berlin: Springer.
  • A. D. R. Choudary, Constantin Niculescu: Real Analysis on Intervals. Springer, 2014, ISBN 9788132221487, pp. 59-62
  • J. Marshall Ash, Allan Berele, Stefan Catoiu: Plausible and Genuine Extensions of L’Hospital's Rule. Mathematics Magazine, Vol. 85, No. 1 (February 2012), pp. 52–60 (JSTOR)

Notes

  1. Choudary, A. D. R.; Niculescu, Constantin (2014). Real Analysis on Intervals. Springer India. pp. 59–60. ISBN 978-81-322-2147-0.
  2. l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com

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