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Raabe’s criteria
Theorem.
The series $a_{1}\!+\!a_{2}\!+\!a_{3}\!+\cdots$ with positive terms is

convergent if, starting from some value of $n$, its terms fulfil the condition
$\frac{a_{{n+1}}}{a_{n}}\leqq 1\frac{\mu}{n}$ where $\mu$ is a constant and $>1$;

divergent if, starting from some value of $n$, its terms fulfil the condition
$\frac{a_{{n+1}}}{a_{n}}\geqq 1\frac{1}{n}\frac{M}{n^{2}}$ where $M$ is a constant.
Related:
ASeriesRelatedToHarmonicSeries
Type of Math Object:
Theorem
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Reference
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