Difference between revisions of "Raabe criterion"
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| + | {{MSC|46A05}} | ||
{{TEX|done}} | {{TEX|done}} | ||
| − | ''on the convergence of a series of numbers'' | + | ''on the convergence of a series of complex numbers'' |
| − | A series $\ | + | A criterion for the convergence of series of complex numbers $\sum_n a_n$, proved by J. Raabe. If $a_n \neq 0$ and there is a number $R>1$ such that for sufficiently large $n$ the inequality |
\begin{equation} | \begin{equation} | ||
| − | + | \frac{|a_{n+1}|}{|a_n|} \leq 1 - \frac{R}{n} | |
\end{equation} | \end{equation} | ||
| − | + | holds, then $\sum_n a_n$ converges absolutely. If instead there is $R<1$ such that | |
| − | + | \[ | |
| − | + | \frac{|a_{n+1}|}{|a_n|} \geq 1 - \frac{R}{n}\, | |
| + | \] | ||
| + | for sufficiently large $n$, then the series $\sum_n |a_n|$ diverges. However, the series itself might still converge, as can be seen taking | ||
| + | \[ | ||
| + | \sum_n (-1)^n \frac{1}{\sqrt{n}}\, . | ||
| + | \] | ||
| + | Observe moreover that the [[Harmonic series|harmonic series]] $\sum \frac{1}{n}$ (which diverges) and the series $\sum_n \frac{1}{n^2}$ (which converges) have both the property that | ||
| + | \[ | ||
| + | \lim_{n\to \infty} n \left(1-\frac{a_n}{a_{n+1}}\right) = 1\, . | ||
| + | \] | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Kn}}|| K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) | ||
| + | |- | ||
| + | |} | ||
Revision as of 20:59, 9 December 2013
2020 Mathematics Subject Classification: Primary: 46A05 [MSN][ZBL]
on the convergence of a series of complex numbers
A criterion for the convergence of series of complex numbers $\sum_n a_n$, proved by J. Raabe. If $a_n \neq 0$ and there is a number $R>1$ such that for sufficiently large $n$ the inequality \begin{equation} \frac{|a_{n+1}|}{|a_n|} \leq 1 - \frac{R}{n} \end{equation} holds, then $\sum_n a_n$ converges absolutely. If instead there is $R<1$ such that \[ \frac{|a_{n+1}|}{|a_n|} \geq 1 - \frac{R}{n}\, \] for sufficiently large $n$, then the series $\sum_n |a_n|$ diverges. However, the series itself might still converge, as can be seen taking \[ \sum_n (-1)^n \frac{1}{\sqrt{n}}\, . \]
Observe moreover that the harmonic series $\sum \frac{1}{n}$ (which diverges) and the series $\sum_n \frac{1}{n^2}$ (which converges) have both the property that \[ \lim_{n\to \infty} n \left(1-\frac{a_n}{a_{n+1}}\right) = 1\, . \]
References
| [Kn] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
Raabe criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Raabe_criterion&oldid=29180