Stolz–Cesàro theorem

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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

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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

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

 

then

 [1]

Proofs

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Proof of the theorem for the */∞ case

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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

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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

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

Arithmetic mean

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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

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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

 

Examples

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Example 1

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Example 2

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

History

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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

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Statement

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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

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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

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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

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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

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  • 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)
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Notes

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  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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