Series - Radius of Convergence

The radius of convergence is the radius of the largest disk in which the series converges. There are a few methods for finding the radius of convergence. The first condition for convergence is that the last term in the series must go to 0. We could test this by just taking the lim u_n  as n goes to infinity, but this isn't enough to determine convergence.

The easiest test is to make sure that the series converges fast enough and to do that we can use the ratio test.

This limit is the absolute value of the last term in the series over the second last term of series, and if this value is < 1 it converges, if its > 1 it diverges, and if this limit is equal to 0 then the test fails. 

We can use the ratio test to determine the radius of convergence in the following fashion, by equating it to the ratio test to the radius of convergence and solving it.

For the following sum

We just start by plugging into the equation

And this solves to 4 when we take n going to infinity, giving us the radius of convergence. If the absolute value of x is greater than this, it doesn't converge.