 In this video, we wanna find a power series representation for the expression x cubed over x plus two. So I wanna draw to your attention that in our previous video, we actually did something very similar to this, right? If we had the expression y equals one over x plus two, we had saw, we saw in that video that we can find a power series representation of this using the geometric series formula, and this turned out to be n equals zero to infinity. And then we get a negative one to the n over two to the n plus one times x to the n, like we did there. So we can do the power series representation if there wasn't an x cubed in there. But after all, x cubed over x plus one is just x cubed times one over x plus two. So if we insert this power series representation into here, then what we can see is that x cubed over x plus two is gonna equal just x cubed times the thing from above, right, bring it down here, the sum for n equals zero to infinity of negative one to the n over two to the n plus one times x to the n. And so then by the distributive property, if we distribute this x cubed through, that'll just add three to the exponent of the n, in which case then we get the power series representation, we take the sum of n equals zero to infinity, it'll still be alternating negative one to the n over two to the n plus one, and then we get x to the n plus three. And so this gives us now our power series representation. And if you wanna see an expanded form, this would look something like, we're gonna get x cubed over two plus, I guess actually it's minus because it's alternate series, x to the fourth over four plus x to the fifth over eight, we have negative x to the sixth over 16, and then this pattern will continue, continue, continue, continue. And we also saw that in the previous video, the interval of convergence of this power series representation is negative two to two. And that's gonna be the same interval of convergence right here. And this is because x cubed, if we think of it as a power series, right, it is a power series where the coefficient sequence is always zero except from the third spot. Its interval of convergence will be the entire real line. So when we put these two power series together, we take the intersection of the intervals, which just gives us this one right here, negative two to two. And so what's kinda nice is when it comes to, when it comes to a rational function, if we wanna find a power series representation, it turns out the numerator is somewhat irrelevant. If we can find a power series representation for one over f of x, for some polynomial f of x, right, then we can just times that by the polynomial on the top, g of x, and we get a power series representation for any rational function. So what I'm saying is, as long as we know what to do with the denominator, we can handle anything in the numerator. And we'll see in the next video, what we can do when we have more exotic denominator than just a linear factor like x plus two.