6.2.1

Instructor: Suppose that this first power series has an interval of convergence of negative 1, 1. And the second power series has an interval of convergence of negative 2 to 2. The question is, where does the sum-- what is its interval of convergence? 

Well we said earlier, in part a, they have to have a common interval. Well, the common interval here is negative 1 to 1. So that is the interval of convergence of the sum. 

Now, the second one has to do with a composition. First, let's write this equals 0 to infinity of an of 3x to the n. Our composition, the bx to the m there, is 3x. So as long as bx to the m is in the interval negative 1 to 1, then we're in good shape. So what we need to check then, is that the absolute value of 3x is less than 1. 

Well that occurs when the absolute value of x is less than 1/3. All right. So that would mean our interval-- well first our radius is 1/3, which means our interval is negative 1/3 to positive 1/3. Since that's what we were asked for, that is we want there.