Difference between revisions of "Bertrand criterion"
From Encyclopedia of Mathematics
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| − | ''of convergence of series | + | {{TEX|done}} |
| + | |||
| + | ''of convergence of series $\sum_{n=1}^{\infty}a_n$ with positive numbers as terms'' | ||
If | If | ||
| − | + | \begin{equation} | |
| − | + | B_n=\left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n | |
| − | + | \end{equation} | |
and if the limit (finite or infinite) | and if the limit (finite or infinite) | ||
| − | + | \begin{equation} | |
| − | + | B = \lim_{n\to\infty}B_n | |
| − | + | \end{equation} | |
| − | exists, then the series is convergent if | + | exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR></table> | ||
Revision as of 07:38, 13 December 2012
of convergence of series $\sum_{n=1}^{\infty}a_n$ with positive numbers as terms
If \begin{equation} B_n=\left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n \end{equation} and if the limit (finite or infinite) \begin{equation} B = \lim_{n\to\infty}B_n \end{equation} exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand.
References
| [1] | 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=16104
Bertrand criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=16104
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article