Interval of Convergence Calculator

About this calculator

A power series converges only within a certain distance of its center, and that distance is the radius of convergence R, typically found by applying the ratio test to the series' terms — taking the limit of the ratio of consecutive coefficients as the index grows. Once R is known, the series is guaranteed to converge for every x strictly between a minus R and a plus R, and to diverge outside that range.

This calculator builds the open interval (a − R, a + R) from a center a and radius R, which is the day-to-day task in calculus courses covering Taylor and Maclaurin series. What the ratio test cannot settle is what happens exactly at the two endpoints — those require separate tests (such as the alternating series test or a p-series comparison) and are checked individually, which is why textbooks and this tool both flag the endpoints as something to verify on their own.