Ratio-Like and Recurrence Relation Tests for Convergence of Series

Three tests for convergence of series are given which are wellsuited for the automatic determination of the radius of convergenceof a series. The first test is similar to the standard ratiotest, while the other two tests use a recurrence relation forthe terms of the series. Together, these tests determine theradius of convergence of many series which arise as solutionsto ordinary differential equations. Numerical examples usingthese tests are given.