Namespaces
Variants
Actions

Difference between revisions of "Conditional convergence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
Line 1: Line 1:
 
''of a series''
 
''of a series''
  
A property of series, stating that the given series converges after a certain (possibly trivial) rearrangement of its terms. A series of numbers
+
A property of series, stating that the given series converges after a
 +
certain (possibly trivial) rearrangement of its terms. A series of
 +
numbers  
 +
$$\sum_{n=1}^\infty u_n\label{*}$$
 +
is unconditionally convergent if it converges itself, as
 +
well as any series obtained by rearranging its terms, while the sum of
 +
any such series is the same; in other words: The sum of an
 +
unconditionally-convergent series does not depend on the order of its
 +
terms. If the series (*) converges, but not unconditionally, then it
 +
is said to be conditionally convergent. For the series (*) to be
 +
conditionally convergent it is necessary and sufficient that it
 +
converges and does not absolutely converge, i.e. that $\sum_{n=1}^\infty |u_n| = +\infty$.
  
<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/c/c024/c024460/c0244601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
If the terms of the series (*) are real numbers, if the non-negative
 +
terms are denoted by $u_1^+, u_2^+,\dots $ and the negative terms by $-u_1^-, -u_2^-,\dots $ then the series
 +
(*) is conditionally convergent if and only both series $\sum_{n=1}^\infty u_n^+ $ and $\sum_{n=1}^\infty u_n^- $
 +
diverge (here the order of the terms in the series is immaterial).
  
is unconditionally convergent if it converges itself, as well as any series obtained by rearranging its terms, while the sum of any such series is the same; in other words: The sum of an unconditionally-convergent series does not depend on the order of its terms. If the series (*) converges, but not unconditionally, then it is said to be conditionally convergent. For the series (*) to be conditionally convergent it is necessary and sufficient that it converges and does not absolutely converge, i.e. that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244602.png" />.
+
Let the series (*) of real numbers be conditionally convergent and let
 +
$-\infty\le \alpha < \beta \le +\infty$, then there exists a series $\sum_{n=1}^\infty u_n^*$, obtained by rearranging the terms
 +
of (*), such that if $\{s_n^*\}$ denotes its sequence of partial sums, then
  
If the terms of the series (*) are real numbers, if the non-negative terms are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244603.png" /> and the negative terms by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244604.png" /> then the series (*) is conditionally convergent if and only both series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244606.png" /> diverge (here the order of the terms in the series is immaterial).
+
$$\underline{\lim}_{n\to\infty}\; s_n^* = \alpha,\quad \overline{\lim}_{n\to\infty}\; s_n^* = \beta$$
 +
(this is a generalization of Riemann's theorem, cf.
 +
[[Riemann theorem|Riemann theorem]] 2).
  
Let the series (*) of real numbers be conditionally convergent and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244607.png" />, then there exists a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244608.png" />, obtained by rearranging the terms of (*), such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c0244609.png" /> denotes its sequence of partial sums, then
+
The product of two conditionally-convergent series depends on the
 +
order in which the result of the term-by-term multiplication of the
 +
two series is summed.
  
<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/c/c024/c024460/c02446010.png" /></td> </tr></table>
+
The concepts of conditional and unconditional convergence of series
 
+
may be generalized to series with terms in some normed vector space
(this is a generalization of Riemann's theorem, cf. [[Riemann theorem|Riemann theorem]] 2).
+
$X$. If $X$ is a finite-dimensional space then, analogously to the
 
+
case of series of numbers, a convergent series $\sum_{n=1}^\infty u_n$, $u_n\in X$, $n=1,2,\dots$ is
The product of two conditionally-convergent series depends on the order in which the result of the term-by-term multiplication of the two series is summed.
+
conditionally convergent if and only if the series $\sum_{n=1}^\infty ||u_n||_X$ is
 
+
divergent. If, however, $X$ is infinite dimensional, then there exist
The concepts of conditional and unconditional convergence of series may be generalized to series with terms in some normed vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446012.png" /> is a finite-dimensional space then, analogously to the case of series of numbers, a convergent series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446015.png" /> is conditionally convergent if and only if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446016.png" /> is divergent. If, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446017.png" /> is infinite dimensional, then there exist unconditionally-convergent series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024460/c02446018.png" />.
+
unconditionally-convergent series $\sum_{n=1}^\infty ||u_n||_X = +\infty$.
  
  
  
 
====Comments====
 
====Comments====
A very useful reference on convergence and divergence of series with elements in abstract spaces is [[#References|[a1]]].
+
A very useful reference on convergence and divergence
 +
of series with elements in abstract spaces is
 +
[[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''1. Sequence spaces''' , Springer (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD>
 +
<TD valign="top"> J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , '''1. Sequence spaces''' , Springer (1977)</TD>
 +
</TR><TR><TD valign="top">[a2]</TD>
 +
<TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD>
 +
</TR></table>

Revision as of 11:06, 18 November 2011

of a series

A property of series, stating that the given series converges after a certain (possibly trivial) rearrangement of its terms. A series of numbers $$\sum_{n=1}^\infty u_n\label{*}$$ is unconditionally convergent if it converges itself, as well as any series obtained by rearranging its terms, while the sum of any such series is the same; in other words: The sum of an unconditionally-convergent series does not depend on the order of its terms. If the series (*) converges, but not unconditionally, then it is said to be conditionally convergent. For the series (*) to be conditionally convergent it is necessary and sufficient that it converges and does not absolutely converge, i.e. that $\sum_{n=1}^\infty |u_n| = +\infty$.

If the terms of the series (*) are real numbers, if the non-negative terms are denoted by $u_1^+, u_2^+,\dots $ and the negative terms by $-u_1^-, -u_2^-,\dots $ then the series (*) is conditionally convergent if and only both series $\sum_{n=1}^\infty u_n^+ $ and $\sum_{n=1}^\infty u_n^- $ diverge (here the order of the terms in the series is immaterial).

Let the series (*) of real numbers be conditionally convergent and let $-\infty\le \alpha < \beta \le +\infty$, then there exists a series $\sum_{n=1}^\infty u_n^*$, obtained by rearranging the terms of (*), such that if $\{s_n^*\}$ denotes its sequence of partial sums, then

$$\underline{\lim}_{n\to\infty}\; s_n^* = \alpha,\quad \overline{\lim}_{n\to\infty}\; s_n^* = \beta$$ (this is a generalization of Riemann's theorem, cf. Riemann theorem 2).

The product of two conditionally-convergent series depends on the order in which the result of the term-by-term multiplication of the two series is summed.

The concepts of conditional and unconditional convergence of series may be generalized to series with terms in some normed vector space $X$. If $X$ is a finite-dimensional space then, analogously to the case of series of numbers, a convergent series $\sum_{n=1}^\infty u_n$, $u_n\in X$, $n=1,2,\dots$ is conditionally convergent if and only if the series $\sum_{n=1}^\infty ||u_n||_X$ is divergent. If, however, $X$ is infinite dimensional, then there exist unconditionally-convergent series $\sum_{n=1}^\infty ||u_n||_X = +\infty$.


Comments

A very useful reference on convergence and divergence of series with elements in abstract spaces is [a1].

References

[a1] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108
How to Cite This Entry:
Conditional convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_convergence&oldid=19008
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article