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Raabe criterion

From Encyclopedia of Mathematics
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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=12025
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article