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If for a series of numbers,
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{{MSC|40A05}}
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{{TEX|done}}
  
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An elementary criterion to test the convergence of a series of real numbers, established by [[DAlembert|J. d'Alembert]] in 1768,
 +
and which is also known as ratio test. Consider such a series $\sum_n a_n$ and assume that $a_n \neq 0$.  
 +
* If
 +
\[
 +
\limsup_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} < 1
 +
\]
 +
then the series converges absolutely
 +
* If
 +
\[
 +
\liminf_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} > 1\, ,
 +
\]
 +
then the series diverges.
  
there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300303.png" />, such that, from a certain term onwards, the inequality
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None of the conclusions can be extended to the equality case. In particular for both
 +
the [[Harmonic series|harmonic series]]
 +
\[
 +
\sum_{n=1}^\infty \frac{1}{n}
 +
\]
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(which diverges) and the series
 +
\[
 +
\sum_{n=1}^\infty \frac{1}{n^2}\,
 +
\]
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(which converges) one has
 +
\[
 +
\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = 1\, .
 +
\]
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A slight modification of these two examples provide also the following conclusions
  
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'''Example 1'''
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Let $\{N_k\}$ be an increasing sequence of natural numbers such that
 +
\[
 +
\sum_{n=N_k+1}^{N_{k+1}} \frac{1}{n} \geq 2^k\,
 +
\]
 +
and set
 +
\[
 +
a_n := \frac{1}{2^k n} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, .
 +
\]
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It is easy to check that the series $\sum a_n$ diverges and
 +
\[
 +
\liminf_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = \frac{1}{2} < 1 = \limsup_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}\, .
 +
\]
  
is satisfied, the series converges absolutely; if, from a certain term onwards,
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'''Example 2'''
 
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Let $\{N_k\}$ be an increasing sequence of natural numbers such that
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300305.png" /></td> </tr></table>
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\[
 
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\sum_{n>N_k}^\infty \frac{1}{n^2} < 3^{-n}\,  
the series diverges. In particular, if the limit
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\]
 
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and set
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300306.png" /></td> </tr></table>
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\[
 
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a_n = \frac{2^k}{n^2} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, .
exists, the series converges absolutely, and if
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\]
 
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It is easy to check that the series $\sum a_n$ converges and
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300307.png" /></td> </tr></table>
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\[
 
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\liminf_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 1 < \limsup_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 2\, .
it diverges. For example, the series
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\]
 
 
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converges absolutely for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d0300309.png" />, since
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d03003010.png" /></td> </tr></table>
 
 
 
while the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d03003011.png" /> diverges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d03003012.png" /> since
 
 
 
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If
 
 
 
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the series may converge or diverge; this condition is satisfied by the two series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030030/d03003015.png" /></td> </tr></table>
 
 
 
the first series being convergent, while the second is divergent.
 
 
 
Established by J. d'Alembert (1768).
 
 
 
 
 
 
 
====Comments====
 
This criterion also goes by the name of ratio test, cf. [[#References|[a1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
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* {{Ref|Ru}} W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976) {{ZBL|0346.26002}}

Latest revision as of 11:28, 22 March 2023

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

An elementary criterion to test the convergence of a series of real numbers, established by J. d'Alembert in 1768, and which is also known as ratio test. Consider such a series $\sum_n a_n$ and assume that $a_n \neq 0$.

  • If

\[ \limsup_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} < 1 \] then the series converges absolutely

  • If

\[ \liminf_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} > 1\, , \] then the series diverges.

None of the conclusions can be extended to the equality case. In particular for both the harmonic series \[ \sum_{n=1}^\infty \frac{1}{n} \] (which diverges) and the series \[ \sum_{n=1}^\infty \frac{1}{n^2}\, \] (which converges) one has \[ \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = 1\, . \] A slight modification of these two examples provide also the following conclusions

Example 1 Let $\{N_k\}$ be an increasing sequence of natural numbers such that \[ \sum_{n=N_k+1}^{N_{k+1}} \frac{1}{n} \geq 2^k\, \] and set \[ a_n := \frac{1}{2^k n} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, . \] It is easy to check that the series $\sum a_n$ diverges and \[ \liminf_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = \frac{1}{2} < 1 = \limsup_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}\, . \]

Example 2 Let $\{N_k\}$ be an increasing sequence of natural numbers such that \[ \sum_{n>N_k}^\infty \frac{1}{n^2} < 3^{-n}\, \] and set \[ a_n = \frac{2^k}{n^2} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, . \] It is easy to check that the series $\sum a_n$ converges and \[ \liminf_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 1 < \limsup_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 2\, . \]

References

  • [Ru] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) Zbl 0346.26002
How to Cite This Entry:
D'Alembert criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_criterion_(convergence_of_series)&oldid=14634
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article