Calculus: the limit ratio test

Tutoring calculus, series convergence comes up. The tutor mentions the limit ratio test for convergence of an infinite series.

Put concisely, the limit ratio test looks at the following: lim_n->∞|t_n+1 ÷ t_n|. If the limit < 1, then the series converges; if it's = 1, the test is inconclusive. If said limit > 1, then the series diverges.

The limit ratio test is very useful, especially for factorials and other such awkward expressions. A quick example of the limit ratio test:

Σn!/(n2ⁿ) (terms from n=0 to ∞) diverges because

|(n+1)!/((n+1)2ⁿ⁺¹) ÷ n!/(n2ⁿ)| = |n!/2ⁿ⁺¹ X 2ⁿ/(n-1)!|

Which simplifies to

|n/2|, the limit of which, as n-> ∞, is > 1.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, part one, third edition. Toronto: D. C. Heath and Company, 1989.