#1716 · Real Analysis · Series

Higher-order ratio test

If $\frac{u _{n}}{u _{n + 1}} = \frac{n ^{k} + A n ^{k - 1} + B n ^{k - 2} + C n ^{k - 3} + im es im es im es}{n ^{k} + a n ^{k - 1} + b n ^{k - 2} + c n ^{k - 3} + im es im es im es}$ , where $k$ is a positive integer, show that the series $u_{1} + u_{2} + u_{3} + im es im es im es$ is convergent if $A - a - 1$ is positive, and divergent if $A - a - 1$ is negative or zero.

~25 minProof · checked by AIRaabe's testlogarithmic test

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXI. Convergency and Divergency of Series, Examples XXI.b, No. 12

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