Geometric series as a function

We have a function right over here defined as an infinite series. What I want to attempt to do in this video is to see if we can express it in a more traditional form. And so the thing that might jump out at you is, well, look, if this is a geometric series, we know how to take the sum of an infinite geometric series, at least over the x values where the thing will actually converge. And so let's first see if this is a geometric series. Well, the telltale pattern of a geometric series is when you go from one term to the next, you're multiplying by a common ratio. So let's see. To go from 2 to negative 8x squared, what do you have to multiply by? We have to multiply by negative 4x squared. So we've multiplied by negative 4x squared. Now, do we multiply by that same amount to get to 32x to the fourth? Well sure. Negative 4x squared times negative 8x squared is going to be positive 32x to the fourth. Multiply by negative 4x squared again, you're going to get to negative 128x to the sixth. So the common ratio is negative 4x squared, our first term is 2, so we can rewrite this. We can rewrite f of x as being equal to the sum from n equals 0 to infinity of-- let's see, our first term here is 2-- 2 times negative 4x squared to the n-th power. This is a geometric series where our common ratio is negative 4x squared to the n-th power. Now, when will this thing right over here converge? Well, we know that a geometric series will converge if the absolute value of its common ratio is less than 1. So let me write this down. So converge if the absolute value of the common ratio, negative 4x squared, is less than 1. Well, this-- the way it's written right now-- is going to be a negative value. So the absolute value of this is just going to be 4x squared. Right? x squared is going to be non-negative, so 4x squared is going to be non-negative. Negative 4x squared is going to be non-positive. So if you're taking the absolute value of a non-positive thing, that's going to be the same thing as the absolute value of the negative of it. So this just has to be less than 1. And the absolute value of something that is strictly non-negative like this, well, that's just going to be 4x squared-- these two statements are equivalent-- and that has to be less than 1. Can divide both sides by 4, you get x squared is less than 1/4. And so we can say that the absolute value of x has to be less than 1/4, or we could say that negative 1/4 has to be less than x, which has to be less than positive 1/4. So expressed this way, we're giving the interval of convergence. This thing will converge as long as x is in this interval. Expressed this way, we're really saying the radius of convergence. This will converge as long as x is less than our radius of convergence, as long as the absolute value of x is less than our radius of convergence, as long as x stays less than 1/4 away from 0. To make it a little bit clearer, you could rewrite this as the distance between x and 0, as long as this- this, you could view this as the distance between x and 0-- as long as this stays less than 1/4, this thing is going to converge. So this is the interval of convergence, this, you could view, 1/4, you could view as the radius of convergence. Now with that out of the way, we've thought about where this thing converges, let's think about what it converges to. Well, we've done this multiple times. This is going to be equal to our first term, 2 over 1 minus our common ratio. Our common ratio is negative 4x squared. So this is going to give us-- and we deserve a drum roll-- 2 over 1 plus 4x squared. So this expression is going to be equal to this, as long as x is within our interval of convergence.