Kronecker's theorem on series convergence

The set of Kronecker about series convergence is a tenet of Analysis . It was introduced by Leopold Kronecker (1823-1891) in 1886 and provides a criterion for the convergence of infinite series .

Formulation of the sentence

Let any sequence of real numbers be given . Then there is a necessary and sufficient condition for the series to converge ${\displaystyle (a_{n})_{n\in {\mathbb {N} _{0}}}}$

${\displaystyle \sum _{n=0}^{\infty }{a_{n}}}$,

that for every sequence of positive real numbers that increases monotonically towards , the derived quotient sequence ${\displaystyle (p_{n})_{n\in {\mathbb {N} _{0}}}}$ ${\displaystyle +\infty }$

${\displaystyle q_{n}={\frac {p_{0}\cdot a_{0}+p_{1}\cdot a_{1}+\cdots +p_{n}\cdot a_{n}}{p_{n}}}}$

represents a null sequence .

Inferences

The above sentence immediately leads to the following statement, which is also cited under the name Lemma von Kronecker . For any sequence of real numbers such that ${\displaystyle (a_{n})_{n\in {\mathbb {N} }}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n}}}$

converges, holds

${\displaystyle \lim _{n\to \infty }{{\frac {1}{n}}\cdot (a_{1}+\cdots +a_{n})}=0}$.

From Kronecker's lemma, by setting for immediate, it follows that the harmonic series must be divergent. ${\displaystyle a_{n}=1}$ ${\displaystyle n\in {\mathbb {N} }}$

In the proof of Kolmogoroff's law of large numbers , Kronecker's lemma provides the decisive argument.

literature

Original work

• Leopold Kronecker: Quelques remarques sur la détermination des Valeurs moyennes . In: Comptes rendus de séances de l'Académie des Sciences de Paris . 103, 1886, pp. 980-987.
• Kurt Hensel (ed.): Leopold Kronecker's works. Published by K. Hensel at the instigation of the Royal Prussian Academy of Sciences. Reprint of the Leipzig 1930 edition. tape 5 . Chelsea Publishing Company, New York, NY 1968.

Monographs

• Paul R. Halmos : Measure Theory . Reprint of the ed. Published by Van Nostrand. Springer-Verlag, New York [u. a.] 1974, ISBN 0-387-90088-8 .
• Konrad Knopp : Theory and Application of Infinite Series . 5th, corrected edition. Springer-Verlag, Berlin [a. a.] 1964, ISBN 3-540-03138-3 .
• Klaus D. Schmidt: Measure and Probability . Springer-Verlag, Berlin [a. a.] 2009, ISBN 978-3-540-89729-3 .

Individual evidence

1. ^ Kronecker: Quelques remarques sur la détermination des Valeurs moyennes . In: Comptes rendus . tape 103 , 1886, p. 980 ff .
2. ^ Kronecker: Quelques remarques sur la détermination des Valeurs moyennes . In: Leopold Kronecker's works . tape V , 1930, p. 301 ff . ( archive.org ).
3. ↑ ^{a b} Knopp: pp. 131, 151.
4. ^ Schmidt: p. 345.
5. ↑ Halmos: pp. 202-204.
6. ^ Schmidt: pp. 345-346.