Theorem of Olivier

The set of Olivier is a mathematical theorem of analysis , which is based on a work of the mathematician Louis Olivier in the second volume of Crelle's Journal goes back to the year 1827th The theorem gives a necessary condition for the convergence of series , the terms of which form a monotonically decreasing sequence of positive real numbers , and thereby provides a tightening of the zero sequence criterion . One of the direct application of the theorem is the divergence of the harmonic series.

formulation

Olivier's theorem can be formulated as follows:

Let be a monotonically decreasing sequence of nonnegative real numbers and let the corresponding series converge , so ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a_{1}+a_{2}+\ldots }$

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

Then applies

${\displaystyle \lim _{n\to \infty }{n\cdot a_{n}}=0}$,

that is, the sequence of numbers is a zero sequence . ${\displaystyle (n\cdot a_{n})_{n\in \mathbb {N} }}$

Proof according to Konrad Knopp

The approach to the proof of Olivier's theorem results from the Cauchy criterion for series .

If an arbitrary one is given, one first sets and finds a lower bound , so that for arbitrary with always the inequality ${\displaystyle \varepsilon >0}$ ${\displaystyle {\varepsilon }_{0}={\tfrac {\varepsilon }{2}}}$ ${\displaystyle N=N({\varepsilon }_{0})}$ ${\displaystyle n\;,\;p\in \mathbb {N} }$ ${\displaystyle n\;,\;p\geq N}$

${\displaystyle a_{p+1}+a_{p+2}+\ldots +a_{p+n}<{\varepsilon }_{0}}$

applies.

Because of the presupposed monotony property of the sequence of numbers, this is first

${\displaystyle n\cdot a_{p+n}<{\varepsilon }_{0}}$

and consequently

${\displaystyle 2n\cdot a_{p+n}<\varepsilon }$

given.

But that means in particular that one is always with ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n\geq N}$

${\displaystyle 2n\cdot a_{N+n}<\varepsilon }$

and thus

${\displaystyle (N+n)\cdot a_{N+n}<\varepsilon }$

Has.

You now choose to as the lower bound  . ${\displaystyle \varepsilon }$ ${\displaystyle N(\varepsilon )=2N}$

This results in the inequality for all with paths and ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n\geq 2N}$ ${\displaystyle n-N\geq N}$ ${\displaystyle n=N+(n-N)}$

${\displaystyle n\cdot a_{n}=(N+(n-N))\cdot a_{N+(n-N)}<\varepsilon }$  .

Hence it is a null sequence. ${\displaystyle (n\cdot a_{n})_{n\in \mathbb {N} }}$

annotation

• For

${\displaystyle a_{n}={\frac {1}{n}}\;\;(n\in \mathbb {N} )}$

one has

${\displaystyle \lim _{n\to \infty }{n\cdot a_{n}}=1}$  ,

which implies with Olivier's theorem the divergence of the harmonic series .

• Using the Abelian series , which

${\displaystyle a_{n}={\frac {1}{n\cdot \ln(n)}}\;\;(n\in \mathbb {N} \;,\;n\geq 2)}$

as a general term, one can see that Olivier's theorem only formulates a necessary, but not a sufficient condition . Because the Abelian series is based on a monotonically falling sequence of terms and is there

${\displaystyle \lim _{n\to \infty }{n\cdot a_{n}}=\lim _{n\to \infty }{\frac {1}{\ln(n)}}=0}$  ,

but nevertheless follows with Cauchy's condensation criterion

${\displaystyle \sum _{n=2}^{\infty }{\frac {1}{n\cdot \ln(n)}}=\infty }$

literature

• Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin / Göttingen / Heidelberg / New York 1964, ISBN 3-540-03138-3 ( MR0183997 ).
• Herbert Meschkowski : Infinite rows . 2nd, improved and enlarged edition. BI Wissenschaftsverlag, Mannheim u. a. 1982, ISBN 3-411-01613-2 ( MR0671586 ).
• Louis Olivier: Remarques sur les séries infinies et leur convergence . In: Journal for pure and applied mathematics . tape 2 , 1827, p. 31-44 ( uni-goettingen.de ).

References and comments

1. ↑ Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin / Göttingen / Heidelberg / New York 1964, ISBN 3-540-03138-3 , p. 125-126 ( MR0183997 ).
2. ↑ Herbert Meschkowski : Infinite rows . 2nd, improved and enlarged edition. BI Wissenschaftsverlag, Mannheim u. a. 1982, ISBN 3-411-01613-2 , pp. 28-29 ( MR0671586 ).
3. ^ A. Ostrowski: Complex Function Theory . In: Collected Mathematical Papers , Vol. 5 XIII, Birkhäuser-Verlag, 1984, ISBN 3-7643-1510-5 , p. 163; there this statement is referred to as Olivier's sentence
4. ↑ with the formal setting of ${\displaystyle a_{1}=0}$
5. ↑ Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin / Göttingen / Heidelberg / New York 1964, ISBN 3-540-03138-3 , p. 121, 124 ( MR0183997 ).
6. ↑ Herbert Meschkowski : Infinite rows . 2nd, improved and enlarged edition. BI Wissenschaftsverlag, Mannheim u. a. 1982, ISBN 3-411-01613-2 , pp. 26-27 ( MR0671586 ).