Formula for evaluation of limits of some real-valued sequences
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
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:
![{\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l.\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/79e5161eb1de2ee29c46f37dcec4b736ba3d35fe)
Then, the limit
![{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0db443977afee3ef0c73b7d0406c5d9bbff95e)
Statement of the theorem for the 0/0 case
Let
and
be two sequences of real numbers. Assume now that
and
while
is strictly decreasing. If
![{\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l,\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f11ac2cec4ccefe5112c5412d09f2b03ff6f1ed)
then
[1]
Proofs
Proof of the theorem for the */∞ case
Case 1: suppose
strictly increasing and divergent to
, and
. By hypothesis, we have that for all
there exists
such that
![{\displaystyle \left|\,{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}-l\,\right|<{\frac {\epsilon }{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea0a2c6efd951add2e031fbb72c054bb4ca73e8)
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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7568e49dc800b62b75ba2cf4d30783a91a5208)
Since
is strictly increasing,
, and the following holds
.
Next we notice that
![{\displaystyle a_{n}=[(a_{n}-a_{n-1})+\dots +(a_{\nu +2}-a_{\nu +1})]+a_{\nu +1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86839b7008fea992a4becfad4fa6c65383cf6b8e)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adbcd6cccc93f546ddc22dd04895324e35a6aa72)
Now, since
as
, there is an
such that
for all
, and we can divide the two inequalities by
for all
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16dcdb7b962c6cf90d6273fd089f7ebcbf699f64)
The two sequences (which are only defined for
as there could be an
such that
)
![{\displaystyle c_{n}^{\pm }:={\frac {a_{\nu +1}-b_{\nu +1}(l\pm \epsilon /2)}{b_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb085de91ac3f2b87575a6f406c0ee3a20de72d)
are infinitesimal since
and the numerator is a constant number, hence for all
there exists
, such that
![{\displaystyle {\begin{aligned}&|c_{n}^{+}|<\epsilon /2,\quad \forall n>n_{+},\\&|c_{n}^{-}|<\epsilon /2,\quad \forall n>n_{-},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3208f849e99cd67efdaebfdab6563ebfbaaa0cea)
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/826c2e86be9b573f0869731819916da1c5d32d4b)
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
![{\displaystyle {\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}>2M.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19e80c9b538b38034c99bbcda01353d08db1d373)
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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/355b51759e2fbae06003090086439276f6fd928e)
and
![{\displaystyle {\frac {a_{n}}{b_{n}}}>2M+{\frac {a_{\nu +1}-2Mb_{\nu +1}}{b_{n}}},\quad \forall n>\max\{\nu ,n_{0}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/672ed9c0721f5dbfe95860f34d91171e8b0ceee2)
The sequence
defined by
![{\displaystyle c_{n}:={\frac {a_{\nu +1}-2Mb_{\nu +1}}{b_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b99befbb829ec0da245216d3c5468a1453852727)
is infinitesimal, thus
![{\displaystyle \forall M>0\,\exists {\bar {n}}>n_{0}>0{\text{ such that }}-M<c_{n}<M,\,\forall n>{\bar {n}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f111cf62b6434ab16262bfff4f948fa96c9849e)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d56c0a1976584cf6a019123ebabbee1780291bd8)
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
Case 1: we first consider the case with
and
strictly decreasing. This time, for each
, we can write
![{\displaystyle a_{n}=(a_{n}-a_{n+1})+\dots +(a_{n+\nu -1}-a_{n+\nu })+a_{n+\nu },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4105f9c13e523f14249c550a1d6bce93645f828)
and for any
such that for all
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8371be203fe5f7452b348c993ffcb2bbd2e63c8)
The two sequences
![{\displaystyle c_{\nu }^{\pm }:={\frac {a_{n+\nu }-b_{n+\nu }(l\pm \epsilon /2)}{b_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5a0c46a07a2773b3db40bdaf54320473339eca)
are infinitesimal since by hypothesis
as
, thus for all
there are
such that
![{\displaystyle {\begin{aligned}&|c_{\nu }^{+}|<\epsilon /2,\quad \forall \nu >\nu _{+},\\&|c_{\nu }^{-}|<\epsilon /2,\quad \forall \nu >\nu _{-},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417dd4bf1eab1657ca2b8ed752ebcc6e056e612a)
thus, choosing
appropriately (which is to say, taking the limit with respect to
) 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a912f06df982b3269157bc3c646a458079a1410a)
which concludes the proof.
Case 2: we assume
and
strictly decreasing. For all
there exists
such that for all
![{\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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01a629b1f14c769acbb6d75ba62b9d0585b97433)
Therefore, for each
![{\displaystyle {\frac {a_{n}}{b_{n}}}>2M+{\frac {a_{n+\nu }-2Mb_{n+\nu }}{b_{n}}},\quad \forall n>n_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a362c8055c75ad420624b7ac9f4735897c34e8f5)
The sequence
![{\displaystyle c_{\nu }:={\frac {a_{n+\nu }-2Mb_{n+\nu }}{b_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/031b3eb856f8ba280caf14478b52e7f02d48396c)
converges to
(keeping
fixed). Hence
such that ![{\displaystyle -M<c_{\nu }<M,\,\forall \nu >{\bar {\nu }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5935e5e9f7517d24bd89bf2b860fadac7fa0a9a5)
and, choosing
conveniently, we conclude the proof
![{\displaystyle {\frac {a_{n}}{b_{n}}}>2M+c_{\nu }>M,\quad \forall n>n_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2c4836d33d06ae850b3a4b7632cb94f5ec9feb)
Applications and examples
The theorem concerning the ∞/∞ case has a few notable consequences which are useful in the computation of limits.
Arithmetic mean
Let
be a sequence of real numbers which converges to
, define
![{\displaystyle a_{n}:=\sum _{m=1}^{n}x_{m}=x_{1}+\dots +x_{n},\quad b_{n}:=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/535a6e1db93bd810943298b9be4f884a9a262be7)
then
is strictly increasing and diverges to
. 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3096dc3510c159470e0faaa4987defae0f730d9b)
therefore
![{\displaystyle \lim _{n\to \infty }{\frac {x_{1}+\dots +x_{n}}{n}}=\lim _{n\to \infty }x_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d610bb2a23415b5dea4a98350261c3c3147b230a)
Given any sequence
of real numbers, suppose that
![{\displaystyle \lim _{n\to \infty }x_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e68667b4e113488d30c2e1e63bf0ddf238a4ce)
exists (finite or infinite), then
![{\displaystyle \lim _{n\to \infty }{\frac {x_{1}+\dots +x_{n}}{n}}=\lim _{n\to \infty }x_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d610bb2a23415b5dea4a98350261c3c3147b230a)
Geometric mean
Let
be a sequence of positive real numbers converging to
and define
![{\displaystyle a_{n}:=\log(x_{1}\cdots x_{n}),\quad b_{n}:=n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b26f42863836013f3994f7356b39c37185ecb12)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/044be14bb278dc36edd8ca80adced04174bc3ea0)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9bb0e4c023b5d4313c3866836d234cbb663f7)
since the logarithm is both continuous and injective we can conclude that
.
Given any sequence
of (strictly) positive real numbers, suppose that
![{\displaystyle \lim _{n\to \infty }x_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e68667b4e113488d30c2e1e63bf0ddf238a4ce)
exists (finite or infinite), then
![{\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{x_{1}\cdots x_{n}}}=\lim _{n\to \infty }x_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3af736442ea3bead54a7c8a01d3cba5ecc1670)
Suppose we are given a sequence
and we are asked to compute
![{\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{y_{n}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/892731c0f98c82ee08d00d6ecee2eff4b271efd3)
defining
and
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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e9e7a1f16d5facf30a997f4e181c8a6a8ffcea)
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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f8b93c84e5640afc68b11776265bf751233c88)
This last form is usually the most useful to compute limits
Given any sequence
of (strictly) positive real numbers, suppose that
![{\displaystyle \lim _{n\to \infty }{\frac {y_{n+1}}{y_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a529a40779ed45e8e509308b742de50aabb8c799)
exists (finite or infinite), then
![{\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{y_{n}}}=\lim _{n\to \infty }{\frac {y_{n+1}}{y_{n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf05d5b836c44202b146522bea88b0cfc15d7)
Examples
Example 1
![{\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n}}=\lim _{n\to \infty }{\frac {n+1}{n}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d9963f6c528891ed76f6d8bc38e90e3fdf69d03)
Example 2
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7f17dcc792530a3e955abd73d6fe07b5fa48a0)
where we used the representation of
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).
Statement
The general form of the Stolz–Cesàro theorem is the following:[2] If
and
are two sequences such that
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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83de0ffba9858bc7982f2322937717848160d70b)
Proof
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
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e6b2d5155ee3fd63e096c8f16f20c78076d905)
Proof of the equivalent statement
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
![{\displaystyle {\frac {a_{n}}{b_{n}}}<l,\quad \forall n>\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7de7f746b05855b12d2cead13b3699764a2e4b26)
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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d57af3044ebc968dcce2ef64cba01d38a407324)
Because
, we also have
and we can divide by
to get
![{\displaystyle {\frac {A_{n}}{B_{n}}}<{\frac {A_{\nu }-lB_{\nu }}{B_{n}}}+l,\quad \forall n>\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9fa034d93f758fac35cec18be3d21a3dd0d1fd)
Since
as
, the sequence
![{\displaystyle {\frac {A_{\nu }-lB_{\nu }}{B_{n}}}\to 0{\text{ as }}n\to +\infty {\text{ (keeping }}\nu {\text{ fixed)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/561c2bd5584ca09e70bc61b8a4999354c62ae182)
and we obtain
![{\displaystyle \limsup _{n\to \infty }{\frac {A_{n}}{B_{n}}}\leq l,\quad \forall l>L,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1416b52f4f8213276aec33df68ee2a4d74f1e673)
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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9dce7ef993b68cc9b9fc4828291b1bd9d7a9442)
and we are done.
Proof of the original statement
Now, take
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5e6d9e12fa3ad2ea829269a98d47b3ef65f34f)
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
)
![{\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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1a8ff989ab8c6a9a022014d3af606ea29ca919)
which is exactly what we wanted to prove.
References
- 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)
External links
- l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com
- Proof of Stolz–Cesàro theorem at PlanetMath.
Notes
- ^ Choudary, A. D. R.; Niculescu, Constantin (2014). Real Analysis on Intervals. Springer India. pp. 59–60. ISBN 978-81-322-2147-0.
- ^ l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com
This article incorporates material from Stolz-Cesaro theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.