Raabe's criterion

The Raabe criterion or the criterion of Raabe-Duhamel (of Joseph Ludwig Raabe and Jean-Marie Duhamel ) is a mathematical convergence criterion , ie a means of deciding whether a infinite series convergent or divergent.

formulation

1st version

Be an infinite series

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

given with positive real summands that form a monotonically decreasing sequence. ${\displaystyle a_{n}}$

Then is convergent if the consequence ${\displaystyle S}$

${\displaystyle \left(\left({\frac {a_{n+1}}{a_{n}}}-1\right)n\right)_{n\in \mathbb {N} }}$

is limited upwards by a . If all terms of this sequence are greater than , then is divergent. ${\displaystyle -\alpha <-1}$ ${\displaystyle -1}$ ${\displaystyle S}$

2nd version

Be an infinite series

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

given.

Then it is absolutely convergent if for a number almost always (i.e. for ): ${\displaystyle S}$ ${\displaystyle \beta >1}$ ${\displaystyle n\geq n_{0}}$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\leq 1-{\frac {\beta }{n}}}$.

However, it diverges when it almost always fails. ${\displaystyle {\frac {a_{n+1}}{a_{n}}}\geq 1-{\frac {1}{n}}}$

Remarks

As always when considering the convergence behavior of series, this criterion only has to be fulfilled for almost all indices. By changing the criterion of an estimation of results by ${\displaystyle S}$

${\displaystyle T=\sum _{n=0}^{\infty }b_{n}}$

according to the comparison test , wherein the telescopic series with above the zero sequence is. ${\displaystyle T}$ ${\displaystyle b_{n}=c_{n}-c_{n+1}}$ ${\displaystyle c_{n}={\frac {n-1}{\alpha -1}}a_{n}}$

The above results in a series remainder estimate :

${\displaystyle S-S_{N}=\sum _{n=N+1}^{\infty }a_{n}\leq {\frac {N}{\alpha -1}}a_{N+1}}$.

applicability

These criteria are more difficult to apply than the root criterion or quotient criterion , but in uncertain cases often still provide convergence statements. You will e.g. B. used to determine the behavior on the edge of the convergence area in power series .

literature

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