6.2.3

Instructor: What we're going to do is we're going have two infinite series. We're going to multiply them term by term. So in this example we have, we're multiplying 1 over 1 minus x times 1 over 1 minus x squared. To come up with the power series for 1 over 1 minus x times 1 minus x squared on the interval negative 1 to 1. All right. So 1 over 1 minus x is 1 plus x plus x squared plus x cubed plus. And 1 over 1 minus x squared is 1 plus x squared plus x to the fourth, plus x to the sixth, continuing indefinitely. 

So what we are going to do is distribute. Lots of distributing. This is 1 times 1 plus x squared plus x to the fourth plus x to the sixth plus. Plus x times 1 plus x squared plus x to the fourth plus x to the sixth plus dot, dot dot. Plus x squared times x times 1 plus x squared plus x to the fourth plus x to the sixth plus. And we continue. Now if we do a little bit of combining like terms, what you'll notice is there's only going to be one constant term. It's from the very first one. 

I mean one constant term, a 1. There's only one X term. There is only two ways to get an x squared. Two ways to get an x cubed. Three ways to get an x to the fourth, three ways to get an x to the fifth. And that sort of pattern just continues on as long as the absolute value of x is less than 1, because of our interval of convergence right here. Because that's the common interval for these two as well. 

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