ratio test of d’Alembert

A lighter version of the ratio test is the

Ratio test of d’Alembert.  Let  $a_{1}\!+\!a_{2}\!+\ldots$  be a series with positive terms.

$1^{\circ}$.  If there exists a number $q$ such that  $0  and

 $\displaystyle\frac{a_{n+1}}{a_{n}}\;\leq\;q\quad\mbox{for all}\;\;n\geq n_{0},$ (1)

then the series converges.

$2^{\circ}$.  If there exists a number $n_{0}$ such that

 $\displaystyle\frac{a_{n+1}}{a_{n}}\;\geq\;1\quad\mbox{for all}\;\;n\geq n_{0},$ (2)

then the series diverges.

Proof.$1^{\circ}$. By the condition (1), we have  $a_{n+1}\leq a_{n}q$;  thus we get the estimations

 $a_{n_{0}+1}\;\leq\;a_{n_{0}}q,$
 $a_{n_{0}+2}\;\leq\;a_{n_{0}+1}q\;\leq\;a_{n_{0}}q^{2},$
 $\cdots\quad\cdots\quad\cdots$
 $a_{n_{0}+p}\;\leq\;a_{n_{0}+p-1}q\;\leq\;\ldots\;\leq\;a_{n_{0}}q^{p},$
 $\cdots\quad\cdots\quad\cdots$

Because  $a_{n_{0}}q+a_{n_{0}}q^{2}+\ldots+a_{n_{0}}q^{p}+\ldots$  is a convergent geometric series, those inequalities and the comparison test imply that the series

 $a_{n_{0}+1}\!+\!a_{n_{0}+2}\!+\ldots+\!a_{n_{0}+p}\!+\ldots$

and as well the whole series  $a_{1}\!+\!a_{2}\!+\ldots$  is convergent.

$2^{\circ}$.  The condition (2) yields

 $a_{n_{0}+1}\;\geq\;a_{n_{0}},\quad a_{n_{0}+2}\;\geq\;a_{n_{0}+1}\;\geq\;a_{n_% {0}},\quad\ldots$

and since  $a_{n_{0}}$ is positive, the limit of $a_{n}$ as $n$ tends to infinity cannot be 0.  Hence the given series does not fulfil the necessary condition of convergence.

Example.  If the variable $x$ in the power series

 $\sum_{n=0}^{\infty}n!x^{n}$

is distinct from zero, we have

 $\frac{|(n\!+\!1)!x^{n+1}|}{|n!x^{n}|}\;=\;(n\!+\!1)|x|\;\geq\;1\quad\mbox{for % all}\;\;n\geq n_{0}.$

Then the series does not converge absolutely (http://planetmath.org/AbsoluteConvergence).  The known theorem of Abel says that the series diverges for all  $x\neq 0$.  It means that the radius of convergence is 0.

References

• 1 Л. Д. Кудрявцев: Математический анализ. I том.  Издательство  ‘‘Высшая школа’’. Москва (1970).
Title ratio test of d’Alembert RatioTestOfDAlembert 2013-03-22 19:12:28 2013-03-22 19:12:28 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 40A05 FiniteChangesInConvergentSeries