The mth Ratio Test: New Convergence Tests for Series

The famous ratio test of d’Alembert for convergence of series depends on the limit of the simple ratio an+1 an (J. d’Alembert, 1717–1783). If the limit is 1, the test fails. Most notable is its failure in situations where it is expected to succeed. For example, it often fails on series with terms containing factorials or finite products. Such terms appear in Taylor series of many functions. The frequent failure of the ratio test motivated many mathematicians to analyze the ratio an+1 an when its limit is 1. Of course, if the limit of an+1 an is 1, then an+1 an = 1 + bn for some sequence bn that converges to 0. A close look at bn leads to several sharper tests than the ratio test, such as Kummer’s, Raabe’s, and Gauss’s tests. For example, the test which is due to J. L. Raabe (1801–1859) covers some series with factorial terms where the ratio test fails. Some series which are not covered by Raabe’s test can be tested with the sharper test of C. F. Gauss (1777–1855). In fact, Gauss’s test was devised to test the hypergeometric series with unit argument F(α,β; γ ; 1); here