(a) What is the radius of convergence of a power series?
How do you find it?
(b) What is the interval of convergence of a power series?
How do you find it?
a) The radius of convergence is the distance between the endpoints of the radius of convergence and its center. It is found by adding the absolute values of both endpoints together and dividing by two.
b) The interval of convergence is the domain of $x$ values for which the power series converges. It is found by using the ratio test to determine for what values the series is convergent.
that we'LL start with the interval of convergence. The interval of convergence is the largest interval for which your power, Siri's converges and the radius of convergence is just half of the length of that interval. So say that your interval of convergence was something like Minus Are our could be open, Could be closed, could be half open Whatever. Ah, then your radius of convergence would just be R skin that you can find something like this by taking your power. Siri's so it's a look, something like this. If you did the ratio test, then you'd see that you would need for Lim n goes to infinity of absolute value of Ace of N Plus one over Ace of End Times X. That would have to be less than one. Okay, but then this is just going to give you some open interval, and then the case is where you get equal the one the test is inconclusive. So you just need to check the cases where X is equal to one. You know, you just have to check that separately. You can use this to figure out what our is a case of This happens then your R value is limit as n goes to infinity of the absolute value of a n over a n plus one. Okay, And then you know that your interval is gonna look something like minus R T r. Unclear whether or not this should be open or close here. Unclear whether or not this should be open or closed. You just have to check the in points separately. So take minus our plug it into your sum. Figure out whether or not you get convergence or divergence. If you get convergence than you want to include that in your interval of convergence. If you got divergence and you'd leave this part open and similarly, you'd have to check our figure out whether or not that should be included are thrown out.