Gauss's criterion

This article deals with the Gaussian criterion for series convergence. Another use of the term is the Gaussian criterion for quadratic residues in number theory.

The Gaussian criterion is a convergence criterion for series, i.e. a means of deciding whether an infinite series is convergent or divergent . The criterion is also known as the Gauss test for series convergence and is named after Carl Friedrich Gauß .

criteria

Be an infinite series

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

with positive real summands , for whose quotients the following applies: ${\displaystyle a_{n}}$

${\displaystyle {\frac {a_{n+1}}{a_{n}}}=1-{\frac {\alpha }{n}}+{\frac {\theta _{n}}{n^{\lambda }}}}$

or

${\displaystyle {\frac {a_{n}}{a_{n+1}}}=1+{\frac {\alpha }{n}}+{\frac {\tau _{n}}{n^{\lambda }}}}$

with and limited consequences respectively . ${\displaystyle \lambda >1}$ ${\displaystyle (\theta _{n})_{n}}$ ${\displaystyle (\tau _{n})_{n}}$

Then S for convergent, otherwise divergent. ${\displaystyle \alpha >1}$

As always when considering the convergence behavior of series, this criterion only has to be fulfilled for almost all indices.

The criterion of grief can be used as proof .

literature

• Konrad Knopp: Theory and Application of Infinite Series . Springer, 6th edition 1996, ISBN 3-540-59111-7

Web links

• Gaussian criterion at MathWorld
• Further variants of the criterion (DM Bressoud) (PDF file; 127 kB)

Individual evidence

1. ↑ Konrad Knopp: Theory and application of the infinite series , § 38. Springer, 6th edition 1996, ISBN 3-540-59111-7
2. ^ Thomas J. Bromwich: Introduction to the Theory of Infinite Series . AMS 2005, ISBN 978-0821839768