The ratio test is a convergence test for infinite series of positive terms that examines the limit L of the ratio of consecutive terms (a_(n+1)/a_n) as n approaches infinity: the series converges if L < 1, diverges if L > 1, and the test is inconclusive if L = 1. Its justification rests on comparison with a geometric series — bounding the tail of the series by a geometric series with ratio strictly between L and 1 when L < 1 — tying it to the comparison test and the convergence theory of geometric series. This belongs to the calculus topic of infinite series convergence tests, and it underlies later determination of the radius of convergence for Taylor/power series.
The ratio test is a convergence test for infinite series of positive terms that examines the limit L of the ratio of consecutive terms (a_(n+1)/a_n) as n approaches infinity: the series converges if …