 In this video, we're gonna find a power series representation for the function f of x equals one over x plus two. Now the whole basis of this power series representation when you have a rational function is to play off of the geometric series formula. That is the sum as n goes from zero to infinity of r to the n is the same thing as one over one minus r so long as the absolute value of r is less than one. We have a small ratio. So you wanna play off of this observation right here. So what we need to do is somehow or another using algebraic techniques, we need to craft it so that we have a one in the denominator and we need a negative sign right here. And so how can we accomplish that? Well, one way of doing that is in the following way, right? Is you have this one over two plus x. I can get a one in there as from the fact that two is just one plus one. So we get one over one plus one minus x. And then because we have a plus sign here we want a negative sign that actually is gonna tell us that our ratio r is gonna be negative one plus x. Basically I'd use a double negative, double negative going on there. So with that in mind then the geometric series would then be the sum from n equals zero to infinity and we would get negative one plus x raised to the nth power. This is an alternating geometric series n equals zero to infinity. We're gonna get negative one to the n times x plus one to the nth right here. And so we see that if we were to write this in expanded form we're gonna get one minus x plus one. Then we're gonna get plus x plus one squared then minus x plus one cubed and then this pattern will continue on forever. We end up with the power series representation which is centered. It's centered at the value x equals negative one. And if we wanna investigate the interval of convergence we have to take the absolute value of negative x plus one. Well that's just the absolute value of x plus one and that needs to be less than one. This tells us that our radius of convergence is gonna equal one right there. And then to find the interval of convergence we get x plus one needs to be less than one but greater than negative one. That is to say x is less than zero but greater than negative two. So our interval of convergence is right here. The radius convergence is one and we have a power series representation centered at negative one. Now what I wanna mention here is everything we've done is absolutely correct. This is correct but it turns out there there might be a better way of doing this which might seem a little less intuitive when the first time you see it but when we compare the two it'll make more sense. And this is actually how it's recommended in the textbook and this is my recommendation as well. If you have one over two plus x here remember we wanna have a one right here and so we could just take two minus one and get one or we could factor it. Factoring it's gonna be a lot more fruitful for us. If you factor out the two you get one over one plus x over two. And so like we observed above we're gonna take our constant ratio r to be negative x over two. Because again we need a negative sign here so we're using a double negative. And so therefore our series then becomes one half the sum as n equals zero to infinity and we end up with negative x over two to the n. Which we can rewrite that as a sum from n equals zero to infinity and we get negative one to the n it'll be an alternating series x to the n over two to the n plus one. So I'm taking the two n's sorry the n many twos we have in the denominator and then you throw in one more from the factor of one half. So in expanded form this would look a little bit different. You're gonna get one half as your first term. Then your next term you're gonna get minus x over four then you're gonna get plus x squared over eight then you're gonna get minus x cubed over 16 and then this pattern will continue at infinitum. Right it'll just continue on and on and on and on forever. So we do get a different power series representation than we did before. Some things to notice here is that with this power series representation we're now centered we're centered at the origin at x equals zero sorry a equals zero we usually call the center a there a equals zero. So there's one advantage right there that when it comes to working with it we kind of prefer this implicitly being centered at zero but what also happens when we investigate the constant ratio there to find the radius of convergence. So we're gonna take negative x over two this needs to be less than one. Well taking absolute value this will look like one half times the absolute value of x is less than one. So therefore the absolute value of x needs to be less than two. This is our radius of convergence so we can already see sort of a difference here. The radius of convergence and this with this approach where we centered at zero is gonna be twice the radius of convergence than when we centered at negative one. So a bigger radius convergence is actually a good thing. So it means our power series representation will be equal to the one over two plus x more for more values of x. And then we see that the interval of convergence will be x is less than two but greater than negative two which compare that we said when we saw before the previous interval of convergence was from negative two to zero. This one is now from negative two to zero negative two to two excuse me. So this interval of convergence includes the entire interval of convergence from before but also has twice as much. And so this right here will be our ideal power series representation because this will give us the maximal interval of convergence. And this is accomplished by factoring whatever coefficient we see in front of this constant term here. We need a one right there. So whatever it is factor out the coefficient and this will give us a nice big interval of convergence.