Radius and interval of convergence Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.
Here we have to determine the radius of convergence of the .
is a power series .
If the power series converges for |x − c| < R and diverges for |x − c| > R, then 0 ≤ R ≤ ∞ is called the radius of convergence of the power series.
Consider , , let =
we know that this power series will converge for x =0, but that’s it at this point. To determine the remainder of the x’s for which we’ll get convergence we can use any of the tests that we’ve discussed to this point. After application of the test that we choose to work with we will arrive at condition(s) on x that we can use to determine which values of x for which the power series will converge and which values of x for which the power series will diverge. From this we can get the radius of convergence and most of the interval of convergence (with the possible exception of the endpoints).
With all that said, the best tests to use here are almost always the ratio or root test. Most of the power series that we’ll be looking at are set up for one or the other. In this case we’ll use the ratio test.
L = || = ||
= |x+2| (1) = |x+2|
So, the ratio test tells us that if L < 1, then the series will converge, if L > 1, then the series will diverge, and if L = 1, we don’t know what will happen. So, we have,
| < 1 x+2| < 1
That is , |x+2| < series converges.
That is , |x+2| > 1 series diverges.
So, the radius of convergence for this power series is R = 1.