Ermakoff's criterion

The criterion of ermakoff or Ermakoffsche criterion is a mathematical convergence criterion for determining the ( absolute ) convergence and divergence of infinite series , which according to the Russian mathematician Wassili Petrovich ermakoff is titled (1845-1922).

formulation

The function is continuous , positive and monotonically decreasing for . The row has the shape ${\displaystyle f\colon {[0,\infty [}\to \mathbb {R} }$ ${\displaystyle x>1}$

${\displaystyle (*)}$    ${\displaystyle A:=\sum \limits _{n=1}^{\infty }a_{n}=\sum \limits _{n=1}^{\infty }f(n)}$,

where the value of the function defined for is at the point . Then for the series for sufficiently large (e.g. for ) either the inequality for convergence or that for divergence applies: ${\displaystyle f(n)}$ ${\displaystyle x\geqq 1}$ ${\displaystyle f(x)}$ ${\displaystyle x=n}$ ${\displaystyle x}$ ${\displaystyle x\geqq x_{0}}$

${\displaystyle A\colon {\begin{cases}{\text{konvergent, falls}}&{\Bigg .}{\dfrac {f(e^{x})e^{x}}{f(x)}}\leqq q<1{\Bigg .}\\{\text{divergent, falls}}&{\dfrac {f(e^{x})e^{x}}{f(x)}}\geqq 1\,.\end{cases}}}$

proof

The first inequality is satisfied. For each then applies with the substitution : ${\displaystyle x\geqq x_{0}}$ ${\displaystyle t=e^{u}}$

${\displaystyle \int \limits _{e^{x_{0}}}^{e^{x}}f(t)\mathrm {d} t=\int \limits _{x_{0}}^{x}f(e^{u})e^{u}\mathrm {d} u\leqq q\int \limits _{x_{0}}^{x}f(t)\mathrm {d} t}$;

it follows:

${\displaystyle {\begin{aligned}(1-q)\int \limits _{e^{x_{0}}}^{e^{x}}f(t)\mathrm {d} t&\leqq q\left(\,\int \limits _{x_{0}}^{x}f(t)\mathrm {d} t-\int \limits _{e^{x_{0}}}^{e^{x}}f(t)\mathrm {d} t\right)\\&=q\left(\,\int \limits _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t-\int \limits _{x}^{e^{x}}f(t)\mathrm {d} t\right)\leqq q\int \limits _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t\end{aligned}}}$,

because it applies:

${\displaystyle (**)}$    ${\displaystyle e^{x}>x}$,

the subtrahend in the second round bracket is positive. In this case:

${\displaystyle \int \limits _{e^{x_{0}}}^{e^{x}}f(t)\mathrm {d} t\leqq {\frac {1}{1-q}}\int \limits _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t}$;

if we add the integral to both sides , we get: ${\displaystyle \int _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t}$

${\displaystyle \int \limits _{x_{0}}^{e^{x}}f(t)\mathrm {d} t\leqq {\frac {1}{1-q}}\int \limits _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t=L}$

and from this, taking into account : ${\displaystyle (**)}$

${\displaystyle \int \limits _{x_{0}}^{x}f(t)\mathrm {d} t\leqq L\qquad \left(x\geqq x_{0}\right)}$.

Since the integral grows with increasing , it has for a finite limit : ${\displaystyle x}$ ${\displaystyle x\to \infty }$

${\displaystyle \int \limits _{x_{0}}^{\infty }f(t)\mathrm {d} t}$;

according to the integral criterion , the series is thus convergent. ${\displaystyle (*)}$

Now the second inequality is satisfied. Then:

${\displaystyle \int \limits _{e^{x_{0}}}^{e^{x}}f(t)\mathrm {d} t\geqq \int \limits _{x_{0}}^{x}f(t)\mathrm {d} t}$

and, if the integral is added to both sides : ${\displaystyle \int _{x}^{e^{x_{0}}}f(t)\mathrm {d} t}$

${\displaystyle \int \limits _{x}^{e^{x}}f(t)\mathrm {d} t\geqq \int \limits _{x_{0}}^{e^{x_{0}}}f(t)\mathrm {d} t=\gamma >0}$

(because because of is ). Now we form a sequence of numbers by stating ; according to what has been proven: ${\displaystyle (**)}$ ${\displaystyle x_{0}<e^{x_{0}}}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{n-1},x_{n},\ldots }$ ${\displaystyle x_{n}=e^{x_{n-1}}}$

${\displaystyle \int \limits _{x_{n-1}}^{x_{n}}f(t)\mathrm {d} t\geqq \gamma }$,

so:

${\displaystyle \int \limits _{x_{0}}^{x_{n}}f(t)\mathrm {d} t=\sum \limits _{i=1}^{n}\int \limits _{x_{i-1}}^{x_{1}}f(t)\mathrm {d} t\geqq n\gamma }$.

So it is clear that:

${\displaystyle \int \limits _{x_{0}}^{\infty }f(t)\mathrm {d} t=\lim \limits _{x\to \infty }\int \limits _{x_{0}}^{x}f(t)\mathrm {d} t=\infty }$

holds, d. i.e., according to the integral criterion, the series is divergent. ${\displaystyle (*)}$

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. 268/732 p . ( Limited preview in Google Book Search - Russian: Курс дифференциального и интегрального исчисления . Translated by Brigitte Mai, Walter Mai, first edition: 1959).

Individual evidence

1. ↑ Worksheet I. (PDF; 155 kB) Lecture Analysis II (SoSe 2010). Institute for Analysis, Dynamics and Modeling (Faculty of Mathematics and Physics) at the University of Stuttgart, April 29, 2010, p. 3/8 p. , Accessed on December 24, 2012 . 
2. ^ Gregor Michailowitsch Fichtenholz : Differential- und Integralrechner 2 (=  university books for mathematics . Volume 62 ). 10th edition. Verlag Harri Deutsch, Frankfurt am Main 2009, ISBN 978-3-8171-1279-1 , XI: Infinite series with constant terms , p. 268/732 p . ( Limited preview in Google Book Search [accessed December 24, 2012] Russian: Курс дифференциального и интегрального исчисления . Translated by Brigitte Mai, Walter Mai).