Namespaces
Variants
Actions

Difference between revisions of "Bertrand criterion"

From Encyclopedia of Mathematics
Jump to: navigation, search
(+ link)
 
Line 4: Line 4:
 
''for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers''
 
''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
+
A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by [[Joseph Bertrand|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\,  
 
B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\,  

Latest revision as of 10:25, 16 March 2023

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=52676
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article