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.
be two sequences of real numbers.
strictly increasing and divergent to
, 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
strictly decreasing and divergent to
strictly increasing and divergent to
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
appropriately (which is to say, taking the limit with respect to
conveniently, we conclude the proof The theorem concerning the ∞/∞ case has a few notable consequences which are useful in the computation of limits.
be a sequence of real numbers which converges to
is strictly increasing and diverges to
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 Given any sequence
of (strictly) positive real numbers, suppose that exists (finite or infinite), then Suppose we are given a sequence
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 where we used the representation of
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).
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
be any two sequences of real numbers such that then First we notice that: Therefore we need only to show that
, the sequence and we obtain By definition of least upper bound, this precisely means that and we are done.
as in the statement of the general form of the Stolz-Cesàro theorem and define since
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