Set of pride

The Stolz Theorem, Stolz's Limit Theorem or Stolz-Cesàro's Theorem is about limit values in mathematics . It is named after the Austrian mathematician Otto Stolz (1842-1905) and the Italian mathematician Ernesto Cesàro (1859-1906).

sentence

Are and follow in with ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} }$

1. ${\displaystyle \lim a_{n}=\lim b_{n}=0}$and strictly decreasing monotonically or ${\displaystyle b_{n}}$
2. ${\displaystyle \lim b_{n}=\infty }$and growing in a strictly monotonous manner ${\displaystyle b_{n}}$

and the limit exists

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

then applies:

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

Proof of the second case

Following the adoption of convergence of difference quotient with a threshold exists for each one , so all the difference quotient to the index in the area is. So there is one with for each ${\displaystyle c}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N}$ ${\displaystyle k>N}$ ${\displaystyle k}$ ${\displaystyle U_{\varepsilon }(c)}$ ${\displaystyle k}$ ${\displaystyle \eta _{k}}$

${\displaystyle a_{k}-a_{k-1}=(b_{k}-b_{k-1})(c+\eta _{k})}$;

for applies . ${\displaystyle k>N}$ ${\displaystyle |\eta _{k}|<\varepsilon }$

If one adds up these relationships according to from to , one obtains the equation ${\displaystyle k}$ ${\displaystyle N+1}$ ${\displaystyle n\gg N}$

${\displaystyle a_{n}-a_{N}=(b_{n}-b_{N})\,c+\sum _{k=N+1}^{n}(b_{k}-b_{k-1})\,\eta _{k}}$.

Thus applies to the quotient of the terms of the sequence

${\displaystyle {\frac {a_{n}}{b_{n}}}={\frac {a_{N}}{b_{n}}}+\left(1-{\frac {b_{N}}{b_{n}}}\right)c+\sum _{k=N+1}^{n}{\frac {b_{k}-b_{k-1}}{b_{n}}}\,\eta _{k}}$

The first summand on the right-hand side converges to zero, since the sequence grows indefinitely. For the same reason the second term converges to . Because of the monotony of the sequence , the following applies to the third summand ${\displaystyle (b_{n})}$ ${\displaystyle c}$ ${\displaystyle (b_{k})}$

${\displaystyle \left|\sum _{k=N+1}^{n}{\frac {b_{k}-b_{k-1}}{b_{n}}}\,\eta _{k}\right|\leq \sum _{k=N+1}^{n}{\frac {b_{k}-b_{k-1}}{b_{n}}}\,|\eta _{k}|<\left(1-{\frac {b_{N}}{b_{n}}}\right)\varepsilon \leq \varepsilon }$.

One can now find a, so that the difference to the limit value is also limited for all in the first two summands by , for all one then receives the estimate ${\displaystyle M>N}$ ${\displaystyle n>M}$ ${\displaystyle \varepsilon }$ ${\displaystyle n>M}$

${\displaystyle \left|{\frac {a_{n}}{b_{n}}}-c\right|<3\varepsilon }$,

thus the sequence of quotients converges to . ${\displaystyle c}$

To reverse

The reverse of the above sentence is generally wrong. Looking at the two episodes

${\displaystyle (a_{k})=(10,10,100,100,1000,1000,\dotsc )}$

${\displaystyle (b_{k})=(10,11,100,101,1000,1001,\dotsc ),}$

then applies . However, the consequence has no limit. ${\displaystyle {\frac {a_{k}}{b_{k}}}\to 1}$ ${\displaystyle {\frac {a_{k}-a_{k-1}}{b_{k}-b_{k-1}}}}$

generalization

Let two further sequences be given and such that and . Furthermore, let it grow in a strictly monotonous and unlimited manner. ${\displaystyle (r_{n})}$ ${\displaystyle (d_{n})}$ ${\displaystyle \textstyle a_{n}=\sum _{k=1}^{n}r_{k}}$ ${\displaystyle \textstyle b_{n}=\sum _{k=1}^{n}d_{k}}$ ${\displaystyle (b_{n})}$

Out

${\displaystyle {\frac {r_{n}}{d_{n}}}={\frac {a_{n}-a_{n-1}}{b_{n}-b_{n-1}}}\to c}$

then follows

${\displaystyle {\frac {\sum _{k=1}^{n}r_{k}}{\sum _{k=1}^{n}d_{k}}}={\frac {a_{n}}{b_{n}}}\to c}$.

The above requirements are z. B. fulfilled by ${\displaystyle (d_{n})}$

• the harmonic sequence , d. H. , ${\displaystyle d_{n}={\frac {1}{n}}}$ ${\displaystyle b_{n}=H_{n}}$
• any sequence with a positive limit, such as , d. H. , ${\displaystyle d_{n}=1}$ ${\displaystyle b_{n}=n}$
• each monotonically increasing sequence, like , d. H. . ${\displaystyle d_{n}=2n-1}$ ${\displaystyle b_{n}=n^{2}}$

Remarks

A special case is Cauchy's limit theorem , that the sequence of the Cesàro means of a convergent sequence converges again to the limit value of the sequence.

In a sense, Stolz's theorem for calculating limit values ​​for sequences is an analogue to de l'Hospital’s rule for calculating limit values ​​for functions.

literature

• Marian Mureşan: A Concrete Approach to Classical Analysis. Springer, 2008, ISBN 978-0-387-78932-3 , pp. 85-88 ( excerpt (Google) )
• ADR Choudary, Constantin Niculescu: Real Analysis on Intervals. Springer, 2014, ISBN 978-81-322-2148-7 , pp. 59-62 ( excerpt (Google) )
• J. Marshall Ash, Allan Berele, Stefan Catoiu: Plausible and Genuine Extensions of L'Hospital's Rule . In: Mathematics Magazine , Vol. 85, No. 1, February 2012, pp. 52–60, doi: 10.4169 / math.mag.85.1.52 ( JSTOR 10.4169 / math.mag.85.1.52 )

Web links

• Gabriel Nagy: The Stolz-Cesaro Theorem. (PDF)
• Exercise sheet with instructions for proof (PDF; 49 kB)
• L'Hopital's theorem and Cesaro-Stolz's theorem on imomath.com