Difference between revisions of "Bertrand criterion"
From Encyclopedia of Mathematics
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| − | '' | + | ''for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers'' |
| − | + | A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by J. Bertrand. Assume that the limit | |
\begin{equation} | \begin{equation} | ||
| − | + | B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\, | |
\end{equation} | \end{equation} | ||
| − | and if the limit ( | + | exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence cannot be decided, as it is witnessed by the examples |
| − | \ | + | \[ |
| − | + | \sum_{n\geq 2} \frac{1}{n \log n} | |
| − | \ | + | \] |
| − | + | (which diverges) and | |
| + | \[ | ||
| + | \sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\, | ||
| + | \] | ||
| + | (which converges). | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Fi}}|| G.M. Fichtenholz, "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft. (1964) | ||
| + | |- | ||
| + | |} | ||
Revision as of 21:29, 9 December 2013
2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]
for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers
A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by J. Bertrand. Assume that the limit \begin{equation} B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\, \end{equation} exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence cannot be decided, as it is witnessed by the examples \[ \sum_{n\geq 2} \frac{1}{n \log n} \] (which diverges) and \[ \sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\, \] (which converges).
References
| [Fi] | G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964) |
How to Cite This Entry:
Bertrand criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=29179
Bertrand criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=29179
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article