 Welcome back everyone. In our previous video, we saw how we could actually take a geometric series and find a rational function for which it represents. But typically the direction is going to go the other way around. If we start off with a rational function, can we find a geometric series which represents it? And on what values of x does it represent? What is the associated interval of convergence? The interval of convergence will give us where does the rational function equal its power series representation? So we we have to play around with the geometric series formula, right? So we get this a over 1 minus r is equal to our geometric series, right? Which simplifying, simplifying just one thing here, if we start the series at n equals 0, if we start this at n equals 0, then we're going to get that as the sum goes from n equals 0 to infinity, we're going to get r to the n and this will equal 1 over 1 minus r. And so that's this is this is the basic template we're going to use to try to force things to work in our favor right here. So comparing comparing this original rational function with our target rational function here, we see that we want to take this positive x squared and set that equal to negative r. That is to say we need that x squared equals negative r and that's because of the disconnect. We have a positive here and a minus there or in other words, we should say that r is equal to negative x squared. And so we make that substitution. With that substitution in mind, we see that 1 plus 1 over 1 plus x squared is equal to the geometric series starting at n equals 0, unless you have a strong reason to do otherwise start your geometric series at 0. We're going to take the sum from n equals 0 to infinity and we replace the r with a negative x squared and then this will be raised to the nth power. And that's that's all really have to get here and in which case we can simplify it a little bit n equals 0 to infinity of the sum. We're going to get negative 1 to the n times x to the 2n. So notice this is going to be an alternating series, an alternating geometric series in expanded form. This would look like 1 minus x squared plus x to the 4th minus x to the 6th plus x to the 4th, 8th minus x to the 10th. And then the pattern just kind of continues on and on and on and on here. Notice what we're getting is we're getting this infinite series where the coefficient is always positive or negative, positive negative, it will alternate positive, negative, positive, negative, positive, negative. And we also only grab the even powers of x. So we don't have any odd powers. So we have x to the 0x squared, x to the 4th, x to the 6th, etc, etc, etc. And so this is the general formula for our geometric series. We can see it right here. But as it sometimes is useful, we can write in an expanded form like this. But even still we still have to figure out where is equality here, where is equality hold. Sometimes we use this equal sign very, very loosely right with functions because these variables x, but equality can only happen when the left side and the left side are both defined. Now the left hand side right here, it's it's defined for any real number x, but this geometric series on the other hand, it's only defined for certain values of x, its interval of convergence. Now because of our ratio being negative x squared, we can see very quickly that we want the absolute value of r, aka negative x squared. We want this to be less than one. Well, since you're taking absolute value, the the negative sign really does nothing here. We need the absolute value of x squared to be less than one. And again, if you're squaring, or if you take the absolute value of a square, that's the same thing as the square of the absolute value in which case be the absolute value of x to be less than one. When we take the square root of both sides, so we still need that x needs to be between negative one and one, or in other words, the interval of convergence is from negative one to one. I do want to mention that when it comes to a geometric series, a geometric power series, we don't actually need to use the ratio test to find the radius of convergence, because it is a geometric series. The radius of convergence is always going to be derived from this observation, the absolute value of r is less than one. And that typically leads to a simpler calculation than what's associated with the ratio test. And so what we've now done is we found a power series representation, it's right here, for our function one over one plus x squared. And we know that this equality will be valid so long as x is between negative one and one.