Girish Mahajan (Editor)

Stolz–Cesàro theorem

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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.

Contents

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 (the ∙/∞ case)

Let ( a n ) n ≥ 1 and ( b n ) n ≥ 1 be two sequences of real numbers. Assume that ( b n ) n ≥ 1 is strictly monotone and divergent sequence (i.e. strictly increasing and approaches + ∞ or strictly decreasing and approaches − ∞ ) and the following limit exists:

lim n → ∞ a n + 1 − a n b n + 1 − b n = ℓ .  

Then, the limit

lim n → ∞ a n b n  

also exists and it is equal to ℓ.

History

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

It appears as Problem 70 in Pólya and Szegő.

The General Form

The general form of the Stolz–Cesàro theorem is the following: If ( a n ) n ≥ 1 and ( b n ) n ≥ 1 are two sequences such that ( b n ) n ≥ 1 is monotone and unbounded, then:

lim inf n → ∞ a n + 1 − a n b n + 1 − b n ≤ lim inf n → ∞ a n b n ≤ lim sup n → ∞ a n b n ≤ lim sup n → ∞ a n + 1 − a n b n + 1 − b n .

References

Stolz–Cesàro theorem Wikipedia
(Text) CC BY-SA


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