Finding Radius of Convergence

Real Analysis · Axiom Academy

EXAMPLE Finding Radius of Convergence Using the Ratio Test to find the radius and interval of convergence of a power series. Find the radius of convergence R and the interval of convergence of the power series Recall: by the Ratio Test the series converges when , which gives . The endpoints |x-a|=R must then be checked separately. Interval of convergence: ( −∞ , ∞ ). Because , the convergence region is the whole real line, so there are no finite endpoints to mark. Nice work. You found the radius and interval of convergence of with the Ratio Test, including the endpoint reasoning. The Ratio Test sets the radius: , and gives with . Factorials beat powers: here , so for every x . Infinite radius: since L=0 no matter what x is, . Endpoints: a finite radius R forces you to test by hand — but has no finite endpoints, so the interval is all of with nothing left to check. Result: and the interval of convergence is . (This series is exactly the Taylor series for e^ x .) The same recipe handles any power series: identify c_n , form , take the limit to get R , then — whenever R is finite — check both endpoints separately to pin down the full interval.

This is the written version of the interactive lesson above. See the full Real Analysis course.