6.1.3

[STARTS AT 7:48]

Instructor: Find the power series for f of x equals x cubed over 2 minus x. 

Let's begin by rearranging that. We'll factor out an x cubed over 2. Because I want it to be a 1 minus something in the denominator, and a 1 in the numerator. So if I factor out the 2, that's 1 minus x over 2. Now again, that looks very much like our geometric formula. So n equals 0 to infinity. I'll leave the x cubed over 2 here. And that would be x over 2 to the n. As long as the absolute value of x over 2 is less than 1, which would imply absolute value of x is less than 2. 

So let's go ahead and rewrite this. That is the sum of n equals 0 to infinity of x to the n plus 3 over 2 to the n plus 1. Again, for absolute value of x less than 2. Now what we are saying, again, is that x cubed over 2 minus x is equal to x cubed over 2 plus x to the fourth over 2 squared plus x to the fifth over 2 cubed plus, et cetera, as long as the absolute value of x is less than 2.