In mathematics, the Stolz–Cesàro 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 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

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

Then, the limit

Statement of the theorem for the 0/0 case

be two sequences of real numbers is strictly decreasing. If

then

Proofs

Proof of the theorem for the */∞ case

Case 1:

which is to say

, and the following holds

1. .

Next we notice that

thus, by applying the above inequality to each of the terms in the square brackets, we obtain

)

are infinitesimal, such that

therefore

is similar.

Case 2:

Again, by applying the above inequality to each of the terms inside the square brackets we obtain

and

defined by

is infinitesimal, thus

combining this inequality with the previous one we conclude

all proceed in this same way.

Proof of the theorem for the 0/0 case

Case 1:, we can write

we have

The two sequences

such that

) we obtain

which concludes the proof.

Case 2:

The sequence

fixed). Hence

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

, define

. We compute

therefore

of real numbers, suppose that

exists (finite or infinite), then

Geometric mean

be a sequence of positive real numbers 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

1. .

of (strictly) positive real numbers, suppose that

exists (finite or infinite), then

and we are asked to compute

we obtain

if we apply the property above

This last form is usually the most useful to compute limits of (strictly) positive real numbers, suppose that

exists (finite or infinite), then

Examples

Example 1

Example 2

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: is monotone and unbounded, then:

Proof

be any sequence, its partial sum. The equivalent statement we shall prove is: 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;

• .

such that

We can use this inequality so as to write

to get

, the sequence

and we obtain

By definition of least upper bound, this precisely means that

and we are done.

Proof of the original statement

as in the statement of the general form of the Stolz-Cesàro theorem and define

)

which is exactly what we wanted to prove.

References

• .

• .

• .

• .

• A. D. R. Choudary, Constantin Niculescu: Real Analysis on Intervals. Springer, 2014, , 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)

External links

• l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com

Notes