Namespaces
Variants
Actions

Difference between revisions of "Bertrand criterion"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (TeX encoding is done)
Line 1: Line 1:
 +
{{MSC|40A05}}
 
{{TEX|done}}
 
{{TEX|done}}
  
−
''of convergence of series $\sum_{n=1}^{\infty}a_n$ with positive numbers as terms''
+
''for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers''
  
−
If
+
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_n=\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\,
 
\end{equation}
 
\end{equation}
−
and if the limit (finite or infinite)
+
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
−
\begin{equation}
+
\[
−
B = \lim_{n\to\infty}B_n
+
\sum_{n\geq 2} \frac{1}{n \log n}
−
\end{equation}
+
\]
−
exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand.
+
(which diverges) and
 +
\[
 +
\sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\,
 +
\]
 +
(which converges).
  
 
====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>
+
{|
 +
|-
 +
|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=30917
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=30917"