Bertrand's criterion

The criterion of Bertrand or Bertrand's criterion is a mathematical convergence criterion for determining the ( absolute ) convergence and divergence of infinite series , which according to the French mathematician Joseph Bertrand is named (1822-1900).

formulation

Let be a positive real sequence and its series. The result with: ${\displaystyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {R} _{+}^{\mathbb {N} }}$ ${\displaystyle \textstyle A:=\sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle (B_{n})_{n\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }}$

${\displaystyle B_{n}:=\ln(n)\left(n\left({\frac {a_{n}}{a_{n+1}}}-1\right)-1\right)}$

have the finite or infinite (or improper) limit value : ${\displaystyle B\in {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,+\infty \}}$

${\displaystyle B:=\lim \limits _{n\to \infty }B_{n}}$.

Then applies to the series: is . ${\displaystyle A}$ ${\displaystyle {\begin{cases}{\text{konvergent, falls}}&B>1\\{\text{divergent, falls}}&B<1\end{cases}}}$

proof

Be with . The series diverges due to the integral criterion . If we set , then applies and is monotonically decreasing and for and . Furthermore: ${\displaystyle c_{n}:=n\ln(n)}$ ${\displaystyle n\in \mathbb {N} _{\geqq 2}}$ ${\displaystyle \sum _{n=2}^{\infty }{\frac {1}{c_{n}}}}$ ${\displaystyle f(x):={\frac {1}{x\ln(x)}}}$ ${\displaystyle {\frac {1}{c_{n}}}=f(n)}$ ${\displaystyle f(x)}$ ${\displaystyle f(x)\to 0}$ ${\displaystyle x\to \infty }$ ${\displaystyle x\geqq 2}$

${\displaystyle \int _{2}^{R}{\frac {1}{x\ln(x)}}\mathrm {d} x=\int _{2}^{R}{\frac {{\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)}{\ln(x)}}\mathrm {d} x=\ln(\ln(R))-\ln(\ln(2)){\xrightarrow {R\to \infty }}\infty }$.

Now put:

${\displaystyle {\begin{aligned}K_{n}&:=c_{n}{\frac {a_{n}}{a_{n+1}}}-c_{n+1}=n\ln(n){\frac {a_{n}}{a_{n+1}}}-(n+1)\ln(n+1)\\&=n\ln(n){\frac {a_{n}}{a_{n+1}}}-n\ln(n+1)-\ln(n+1)\\&=n\ln(n){\frac {a_{n}}{a_{n+1}}}-n\left(\ln \left(1+{\frac {1}{n}}\right)+\ln(n)\right)-\left(\ln \left(1+{\frac {1}{n}}\right)+\ln(n)\right)\\&=n\ln(n){\frac {a_{n}}{a_{n+1}}}-(n+1)\ln \left(1+{\frac {1}{n}}\right)-n\ln(n)-\ln(n)\\&=\ln(n)\left(n{\frac {a_{n}}{a_{n+1}}}-n-1\right)-\ln \left(1+{\frac {1}{n}}\right)^{n+1}\\&=\ln(n)\left(n\left({\frac {a_{n}}{a_{n+1}}}-1\right)-1\right)-\ln \left(1+{\frac {1}{n}}\right)^{n+1}\\&=B_{n}-\ln \left(1+{\frac {1}{n}}\right)^{n+1}\end{aligned}}}$.

With the continuity of the logarithm and the known limit value, it follows for : ${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n+1}=e}$ ${\displaystyle n\to \infty }$

${\displaystyle K=B-\ln(e)=B-1}$,

where and applies. now fulfills the conditions of the criterion of Kummer after construction . From the latter follows for : . ${\displaystyle K,B\in {\overline {\mathbb {R} }}}$ ${\displaystyle K:=\lim _{n\to \infty }K_{n}}$ ${\displaystyle (c_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle A\colon {\begin{cases}{\text{konvergent}}&K>0\Leftrightarrow B>1\\{\text{divergent}}&K<0\Leftrightarrow B<1\end{cases}}}$

literature

• Gregor Michailowitsch Fichtenholz : Differential and integral calculus 2 (=  university books for mathematics . Volume 62 ). 10th edition. Verlag Harri Deutsch [Fismatgis / Физматгиз ], Frankfurt am Main [Moscow] 2009, ISBN 978-3-8171-1279-1 , XI: Infinite series with constant members , p. 262, 732 ( limited preview in Google Book Search - Russian: Курс дифференциального и интегрального исчисления . Translated by Brigitte Mai, Walter Mai, first edition: 1959).

Individual evidence

1. ^ Markus Oster, Nicolai Lang; Christian Barth: Solutions to Worksheet I. (PDF; 155 kB) Lecture Analysis II (SoSe 2009). October 25, 2009, pp. 7/28 pp. , Accessed on December 23, 2012 .