Stolz–Cesàro theorem

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.

Let ${\displaystyle (a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$ be two sequences of real numbers. Assume that ${\displaystyle (b_{n})_{n\geq 1}}$ is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching ${\displaystyle +\infty }$, or strictly decreasing and approaching ${\displaystyle -\infty }$) and the following limit exists:

${\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l.\ }$

Then, the limit

${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }$

Let ${\displaystyle (a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$ be two sequences of real numbers. Assume now that ${\displaystyle (a_{n})\to 0}$ and ${\displaystyle (b_{n})\to 0}$ while ${\displaystyle (b_{n})_{n\geq 1}}$ is strictly decreasing. If

${\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l,\ }$

then

${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }$^[1]

Case 1: suppose ${\displaystyle (b_{n})}$ strictly increasing and divergent to ${\displaystyle +\infty }$, and ${\displaystyle -\infty <l<\infty }$. By hypothesis, we have that for all ${\displaystyle \epsilon /2>0}$ there exists ${\displaystyle \nu >0}$ such that ${\displaystyle \forall n>\nu }$

${\displaystyle \left|\,{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}-l\,\right|<{\frac {\epsilon }{2}},}$

which is to say

${\displaystyle l-\epsilon /2<{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}<l+\epsilon /2,\quad \forall n>\nu .}$

Since ${\displaystyle (b_{n})}$ is strictly increasing, ${\displaystyle b_{n+1}-b_{n}>0}$, and the following holds

${\displaystyle (l-\epsilon /2)(b_{n+1}-b_{n})<a_{n+1}-a_{n}<(l+\epsilon /2)(b_{n+1}-b_{n}),\quad \forall n>\nu }$.

Next we notice that

${\displaystyle a_{n}=[(a_{n}-a_{n-1})+\dots +(a_{\nu +2}-a_{\nu +1})]+a_{\nu +1}}$

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

${\displaystyle {\begin{aligned}&(l-\epsilon /2)(b_{n}-b_{\nu +1})+a_{\nu +1}=(l-\epsilon /2)[(b_{n}-b_{n-1})+\dots +(b_{\nu +2}-b_{\nu +1})]+a_{\nu +1}<a_{n}\\&a_{n}<(l+\epsilon /2)[(b_{n}-b_{n-1})+\dots +(b_{\nu +2}-b_{\nu +1})]+a_{\nu +1}=(l+\epsilon /2)(b_{n}-b_{\nu +1})+a_{\nu +1}.\end{aligned}}}$

Now, since ${\displaystyle b_{n}\to +\infty }$ as ${\displaystyle n\to \infty }$, there is an ${\displaystyle n_{0}>0}$ such that ${\displaystyle b_{n}>0}$ for all ${\displaystyle n>n_{0}}$, and we can divide the two inequalities by ${\displaystyle b_{n}}$ for all ${\displaystyle n>\max\{\nu ,n_{0}\}}$

${\displaystyle (l-\epsilon /2)+{\frac {a_{\nu +1}-b_{\nu +1}(l-\epsilon /2)}{b_{n}}}<{\frac {a_{n}}{b_{n}}}<(l+\epsilon /2)+{\frac {a_{\nu +1}-b_{\nu +1}(l+\epsilon /2)}{b_{n}}}.}$

The two sequences (which are only defined for ${\displaystyle n>n_{0}}$ as there could be an ${\displaystyle N\leq n_{0}}$ such that ${\displaystyle b_{N}=0}$)

${\displaystyle c_{n}^{\pm }:={\frac {a_{\nu +1}-b_{\nu +1}(l\pm \epsilon /2)}{b_{n}}}}$

are infinitesimal since ${\displaystyle b_{n}\to +\infty }$ and the numerator is a constant number, hence for all ${\displaystyle \epsilon /2>0}$ there exists ${\displaystyle n_{\pm }>n_{0}>0}$, such that

${\displaystyle {\begin{aligned}&|c_{n}^{+}|<\epsilon /2,\quad \forall n>n_{+},\\&|c_{n}^{-}|<\epsilon /2,\quad \forall n>n_{-},\end{aligned}}}$

therefore

${\displaystyle l-\epsilon <l-\epsilon /2+c_{n}^{-}<{\frac {a_{n}}{b_{n}}}<l+\epsilon /2+c_{n}^{+}<l+\epsilon ,\quad \forall n>\max \lbrace \nu ,n_{\pm }\rbrace =:N>0,}$

which concludes the proof. The case with ${\displaystyle (b_{n})}$ strictly decreasing and divergent to ${\displaystyle -\infty }$, and ${\displaystyle l<\infty }$ is similar.

Case 2: we assume ${\displaystyle (b_{n})}$ strictly increasing and divergent to ${\displaystyle +\infty }$, and ${\displaystyle l=+\infty }$. Proceeding as before, for all ${\displaystyle 2M>0}$ there exists ${\displaystyle \nu >0}$ such that for all ${\displaystyle n>\nu }$

${\displaystyle {\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}>2M.}$

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

${\displaystyle a_{n}>2M(b_{n}-b_{\nu +1})+a_{\nu +1},\quad \forall n>\nu ,}$

and

${\displaystyle {\frac {a_{n}}{b_{n}}}>2M+{\frac {a_{\nu +1}-2Mb_{\nu +1}}{b_{n}}},\quad \forall n>\max\{\nu ,n_{0}\}.}$

The sequence ${\displaystyle (c_{n})_{n>n_{0}}}$ defined by

${\displaystyle c_{n}:={\frac {a_{\nu +1}-2Mb_{\nu +1}}{b_{n}}}}$

is infinitesimal, thus

${\displaystyle \forall M>0\,\exists {\bar {n}}>n_{0}>0{\text{ such that }}-M<c_{n}<M,\,\forall n>{\bar {n}},}$

combining this inequality with the previous one we conclude

${\displaystyle {\frac {a_{n}}{b_{n}}}>2M+c_{n}>M,\quad \forall n>\max\{\nu ,{\bar {n}}\}=:N.}$

The proofs of the other cases with ${\displaystyle (b_{n})}$ strictly increasing or decreasing and approaching ${\displaystyle +\infty }$ or ${\displaystyle -\infty }$ respectively and ${\displaystyle l=\pm \infty }$ all proceed in this same way.

Case 1: we first consider the case with ${\displaystyle l<\infty }$ and ${\displaystyle (b_{n})}$ strictly decreasing. This time, for each ${\displaystyle \nu >0}$, we can write

${\displaystyle a_{n}=(a_{n}-a_{n+1})+\dots +(a_{n+\nu -1}-a_{n+\nu })+a_{n+\nu },}$

and for any ${\displaystyle \epsilon /2>0,}$ ${\displaystyle \exists n_{0}}$ such that for all ${\displaystyle n>n_{0}}$ we have

${\displaystyle {\begin{aligned}&(l-\epsilon /2)(b_{n}-b_{n+\nu })+a_{n+\nu }=(l-\epsilon /2)[(b_{n}-b_{n+1})+\dots +(b_{n+\nu -1}-b_{n+\nu })]+a_{n+\nu }<a_{n}\\&a_{n}<(l+\epsilon /2)[(b_{n}-b_{n+1})+\dots +(b_{n+\nu -1}-b_{n+\nu })]+a_{n+\nu }=(l+\epsilon /2)(b_{n}-b_{n+\nu })+a_{n+\nu }.\end{aligned}}}$

The two sequences

${\displaystyle c_{\nu }^{\pm }:={\frac {a_{n+\nu }-b_{n+\nu }(l\pm \epsilon /2)}{b_{n}}}}$

are infinitesimal since by hypothesis ${\displaystyle a_{n+\nu },b_{n+\nu }\to 0}$ as ${\displaystyle \nu \to \infty }$, thus for all ${\displaystyle \epsilon /2>0}$ there are ${\displaystyle \nu _{\pm }>0}$ such that

${\displaystyle {\begin{aligned}&|c_{\nu }^{+}|<\epsilon /2,\quad \forall \nu >\nu _{+},\\&|c_{\nu }^{-}|<\epsilon /2,\quad \forall \nu >\nu _{-},\end{aligned}}}$

thus, choosing ${\displaystyle \nu }$ appropriately (which is to say, taking the limit with respect to ${\displaystyle \nu }$) we obtain

${\displaystyle l-\epsilon <l-\epsilon /2+c_{\nu }^{-}<{\frac {a_{n}}{b_{n}}}<l+\epsilon /2+c_{\nu }^{+}<l+\epsilon ,\quad \forall n>n_{0}}$

which concludes the proof.

Case 2: we assume ${\displaystyle l=+\infty }$ and ${\displaystyle (b_{n})}$ strictly decreasing. For all ${\displaystyle 2M>0}$ there exists ${\displaystyle n_{0}>0}$ such that for all ${\displaystyle n>n_{0},}$

${\displaystyle {\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}>2M\implies a_{n}-a_{n+1}>2M(b_{n}-b_{n+1}).}$

Therefore, for each ${\displaystyle \nu >0,}$

${\displaystyle {\frac {a_{n}}{b_{n}}}>2M+{\frac {a_{n+\nu }-2Mb_{n+\nu }}{b_{n}}},\quad \forall n>n_{0}.}$

The sequence

${\displaystyle c_{\nu }:={\frac {a_{n+\nu }-2Mb_{n+\nu }}{b_{n}}}}$

converges to ${\displaystyle 0}$ (keeping ${\displaystyle n}$ fixed). Hence

${\displaystyle \forall M>0\,~\exists {\bar {\nu }}>0}$ such that ${\displaystyle -M<c_{\nu }<M,\,\forall \nu >{\bar {\nu }},}$

and, choosing ${\displaystyle \nu }$ conveniently, we conclude the proof

${\displaystyle {\frac {a_{n}}{b_{n}}}>2M+c_{\nu }>M,\quad \forall n>n_{0}.}$

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

Let ${\displaystyle (x_{n})}$ be a sequence of real numbers which converges to ${\displaystyle l}$, define

${\displaystyle a_{n}:=\sum _{m=1}^{n}x_{m}=x_{1}+\dots +x_{n},\quad b_{n}:=n}$

then ${\displaystyle (b_{n})}$ is strictly increasing and diverges to ${\displaystyle +\infty }$. We compute

${\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=\lim _{n\to \infty }x_{n+1}=\lim _{n\to \infty }x_{n}=l}$

therefore

${\displaystyle \lim _{n\to \infty }{\frac {x_{1}+\dots +x_{n}}{n}}=\lim _{n\to \infty }x_{n}.}$

Given any sequence ${\displaystyle (x_{n})_{n\geq 1}}$ of real numbers, suppose that

${\displaystyle \lim _{n\to \infty }x_{n}}$

exists (finite or infinite), then

${\displaystyle \lim _{n\to \infty }{\frac {x_{1}+\dots +x_{n}}{n}}=\lim _{n\to \infty }x_{n}.}$

Let ${\displaystyle (x_{n})}$ be a sequence of positive real numbers converging to ${\displaystyle l}$ and define

${\displaystyle a_{n}:=\log(x_{1}\cdots x_{n}),\quad b_{n}:=n,}$

again we compute

${\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=\lim _{n\to \infty }\log {\Big (}{\frac {x_{1}\cdots x_{n+1}}{x_{1}\cdots x_{n}}}{\Big )}=\lim _{n\to \infty }\log(x_{n+1})=\lim _{n\to \infty }\log(x_{n})=\log(l),}$

where we used the fact that the logarithm is continuous. Thus

${\displaystyle \lim _{n\to \infty }{\frac {\log(x_{1}\cdots x_{n})}{n}}=\lim _{n\to \infty }\log {\Big (}(x_{1}\cdots x_{n})^{\frac {1}{n}}{\Big )}=\log(l),}$

since the logarithm is both continuous and injective we can conclude that

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{x_{1}\cdots x_{n}}}=\lim _{n\to \infty }x_{n}}$.

Given any sequence ${\displaystyle (x_{n})_{n\geq 1}}$ of (strictly) positive real numbers, suppose that

${\displaystyle \lim _{n\to \infty }x_{n}}$

exists (finite or infinite), then

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{x_{1}\cdots x_{n}}}=\lim _{n\to \infty }x_{n}.}$

Suppose we are given a sequence ${\displaystyle (y_{n})_{n\geq 1}}$ and we are asked to compute

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{y_{n}}},}$

defining ${\displaystyle y_{0}=1}$ and ${\displaystyle x_{n}=y_{n}/y_{n-1}}$ we obtain

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{x_{1}\dots x_{n}}}=\lim _{n\to \infty }{\sqrt[{n}]{\frac {y_{1}\dots y_{n}}{y_{0}\cdot y_{1}\dots y_{n-1}}}}=\lim _{n\to \infty }{\sqrt[{n}]{y_{n}}},}$

if we apply the property above

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{y_{n}}}=\lim _{n\to \infty }x_{n}=\lim _{n\to \infty }{\frac {y_{n}}{y_{n-1}}}.}$

This last form is usually the most useful to compute limits

Given any sequence ${\displaystyle (y_{n})_{n\geq 1}}$ of (strictly) positive real numbers, suppose that

${\displaystyle \lim _{n\to \infty }{\frac {y_{n+1}}{y_{n}}}}$

exists (finite or infinite), then

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{y_{n}}}=\lim _{n\to \infty }{\frac {y_{n+1}}{y_{n}}}.}$

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n}}=\lim _{n\to \infty }{\frac {n+1}{n}}=1.}$

${\displaystyle {\begin{aligned}\lim _{n\to \infty }{\frac {\sqrt[{n}]{n!}}{n}}&=\lim _{n\to \infty }{\frac {(n+1)!(n^{n})}{n!(n+1)^{n+1}}}\\&=\lim _{n\to \infty }{\frac {n^{n}}{(n+1)^{n}}}=\lim _{n\to \infty }{\frac {1}{(1+{\frac {1}{n}})^{n}}}={\frac {1}{e}}\end{aligned}}}$

where we used the representation of ${\displaystyle e}$ as the limit of a sequence.

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 of the Stolz–Cesàro theorem is the following:^[2] If ${\displaystyle (a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$ are two sequences such that ${\displaystyle (b_{n})_{n\geq 1}}$ is monotone and unbounded, then:

${\displaystyle \liminf _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}\leq \liminf _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}.}$

Instead of proving the previous statement, we shall prove a slightly different one; first we introduce a notation: let ${\displaystyle (a_{n})_{n\geq 1}}$ be any sequence, its partial sum will be denoted by ${\displaystyle A_{n}:=\sum _{m\geq 1}^{n}a_{m}}$. The equivalent statement we shall prove is:

Let ${\displaystyle (a_{n})_{n\geq 1},(b_{n})_{\geq 1}}$ be any two sequences of real numbers such that

• ${\displaystyle b_{n}>0,\quad \forall n\in {\mathbb {Z} }_{>0}}$,
• ${\displaystyle \lim _{n\to \infty }B_{n}=+\infty }$,

then

${\displaystyle \liminf _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \liminf _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}.}$

First we notice that:

• ${\displaystyle \liminf _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}}$ holds by definition of limit superior and limit inferior;
• ${\displaystyle \liminf _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \liminf _{n\to \infty }{\frac {A_{n}}{B_{n}}}}$ holds if and only if ${\displaystyle \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}}$ because ${\displaystyle \liminf _{n\to \infty }x_{n}=-\limsup _{n\to \infty }(-x_{n})}$ for any sequence ${\displaystyle (x_{n})_{n\geq 1}}$.

Therefore we need only to show that ${\displaystyle \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}}$. If ${\displaystyle L:=\limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=+\infty }$ there is nothing to prove, hence we can assume ${\displaystyle L<+\infty }$ (it can be either finite or ${\displaystyle -\infty }$). By definition of ${\displaystyle \limsup }$, for all ${\displaystyle l>L}$ there is a natural number ${\displaystyle \nu >0}$ such that

${\displaystyle {\frac {a_{n}}{b_{n}}}<l,\quad \forall n>\nu .}$

We can use this inequality so as to write

${\displaystyle A_{n}=A_{\nu }+a_{\nu +1}+\dots +a_{n}<A_{\nu }+l(B_{n}-B_{\nu }),\quad \forall n>\nu ,}$

Because ${\displaystyle b_{n}>0}$, we also have ${\displaystyle B_{n}>0}$ and we can divide by ${\displaystyle B_{n}}$ to get

${\displaystyle {\frac {A_{n}}{B_{n}}}<{\frac {A_{\nu }-lB_{\nu }}{B_{n}}}+l,\quad \forall n>\nu .}$

Since ${\displaystyle B_{n}\to +\infty }$ as ${\displaystyle n\to +\infty }$, the sequence

${\displaystyle {\frac {A_{\nu }-lB_{\nu }}{B_{n}}}\to 0{\text{ as }}n\to +\infty {\text{ (keeping }}\nu {\text{ fixed)}},}$

and we obtain

${\displaystyle \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq l,\quad \forall l>L,}$

By definition of least upper bound, this precisely means that

${\displaystyle \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq L=\limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}},}$

and we are done.

Now, take ${\displaystyle (a_{n}),(b_{n})}$ as in the statement of the general form of the Stolz-Cesàro theorem and define

${\displaystyle \alpha _{1}=a_{1},\alpha _{k}=a_{k}-a_{k-1},\,\forall k>1\quad \beta _{1}=b_{1},\beta _{k}=b_{k}-b_{k-1}\,\forall k>1}$

since ${\displaystyle (b_{n})}$ is strictly monotone (we can assume strictly increasing for example), ${\displaystyle \beta _{n}>0}$ for all ${\displaystyle n}$ and since ${\displaystyle b_{n}\to +\infty }$ also ${\displaystyle \mathrm {B} _{n}=b_{1}+(b_{2}-b_{1})+\dots +(b_{n}-b_{n-1})=b_{n}\to +\infty }$, thus we can apply the theorem we have just proved to ${\displaystyle (\alpha _{n}),(\beta _{n})}$ (and their partial sums ${\displaystyle (\mathrm {A} _{n}),(\mathrm {B} _{n})}$)

${\displaystyle \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=\limsup _{n\to \infty }{\frac {\mathrm {A} _{n}}{\mathrm {B} _{n}}}\leq \limsup _{n\to \infty }{\frac {\alpha _{n}}{\beta _{n}}}=\limsup _{n\to \infty }{\frac {a_{n}-a_{n-1}}{b_{n}-b_{n-1}}},}$

which is exactly what we wanted to prove.

• 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)

• l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com
• Proof of Stolz–Cesàro theorem at PlanetMath.

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