Difference between revisions of "Leibniz criterion"
From Encyclopedia of Mathematics
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''for convergence of an alternating series'' | ''for convergence of an alternating series'' | ||
If the terms of an alternating series | If the terms of an alternating series | ||
| − | + | \begin{equation} | |
| − | + | \sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0, | |
| − | + | \end{equation} | |
| − | decrease monotonically ( | + | decrease monotonically ($a_n>a_{n+1}$, $n=1,2,\dots$) and tend to zero ( $\lim\limits_{n\to\infty}a_n=0$ ), then the series converges; moreover, a remainder of the series, |
| − | + | \begin{equation} | |
| − | + | \sum_{k=n+1}^{\infty}(-1)^{k+1}a_k, | |
| − | + | \end{equation} | |
has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682. | has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682. | ||
| − | + | ====Examples==== | |
| − | ==== | + | * [[Leibniz_series|Leibniz series]] $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}$. |
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)</TD></TR></table> | ||
Revision as of 09:52, 12 December 2012
for convergence of an alternating series
If the terms of an alternating series \begin{equation} \sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0, \end{equation} decrease monotonically ($a_n>a_{n+1}$, $n=1,2,\dots$) and tend to zero ( $\lim\limits_{n\to\infty}a_n=0$ ), then the series converges; moreover, a remainder of the series, \begin{equation} \sum_{k=n+1}^{\infty}(-1)^{k+1}a_k, \end{equation} has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.
Examples
- Leibniz series $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}$.
References
| [a1] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
How to Cite This Entry:
Leibniz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz_criterion&oldid=29175
Leibniz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz_criterion&oldid=29175
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article