Difference between revisions of "Raabe criterion"
From Encyclopedia of Mathematics
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''on the convergence of a series of numbers'' | ''on the convergence of a series of numbers'' | ||
| − | A series | + | A series $\sum_{n=1}^{\infty}a_n$ converges if for sufficiently large $n$ the inequality |
| − | + | \begin{equation} | |
| − | + | R_n = n\left(\frac{a_n}{a_{n+1}}-1\right)\geq r>1 | |
| − | + | \end{equation} | |
| − | is fulfilled. If | + | is fulfilled. If $R_n\leq 1$ from some $n$ onwards, then the series diverges. |
Proved by J. Raabe | Proved by J. Raabe | ||
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====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 09:57, 13 December 2012
on the convergence of a series of numbers
A series $\sum_{n=1}^{\infty}a_n$ converges if for sufficiently large $n$ the inequality \begin{equation} R_n = n\left(\frac{a_n}{a_{n+1}}-1\right)\geq r>1 \end{equation} is fulfilled. If $R_n\leq 1$ from some $n$ onwards, then the series diverges.
Proved by J. Raabe
References
| [a1] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
How to Cite This Entry:
Raabe criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Raabe_criterion&oldid=29180
Raabe criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Raabe_criterion&oldid=29180
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article