 Consider the infinite series. The geometric series summation formula gives us a formula for the sum of the series. Now, if x is equal to 2, we can substitute and we find... which seems a little contradictory. And it is. And it's useful to remember in mathematics, if you arrive at a contradiction, you did something you shouldn't have. And in this particular case, we shouldn't have let x equal 2. So why can't we let x equals 2? If x equals 2, our series would be... by applying your favorite convergence test, well, how about the ratio test, we find... the series diverges. And this leads to a key principle. Avoid divergent series. And the natural question to ask at this point is, for what values of x will the series converge? So let's think about this. First, we apply your favorite convergence test and find that our series converges for x equal to c. So, for example, if x equals c, we apply the root test to our series and find... Now, we're assuming that our series converges at c, which means that this limit has to be less than or equal to 1. So if we take c prime, whose absolute value is smaller than the absolute value of c, then this limit will be definitely less than 1. And so our new series will definitely converge. So if the absolute value of c prime is less than the absolute value of c, then our series will also converge. And intuitively, this means that if you converge at a point, you converge for smaller values as well. And this suggests the following definition. The power series has a radius of convergence r if the series converges for all x where the absolute value of x is strictly less than r. To understand why it's called a radius, you should take complex analysis. So, for example, let's find the radius of convergence of our familiar power series. We want to find x, so the series converges. Now, since the terms are powers and nothing else, the root test is probably the easiest to apply. We want the series to converge, so we must have our limit strictly less than 1. And if we do a little algebra, we find, so if we apply our favorite convergence test, we find the absolute value of x must be strictly less than 1. So the series has radius of convergence r equal to 1. Now, you might remember that ordinarily your favorite convergence test has undecided cases. And note that in this case, if the absolute value of fx is equal to 1, the root test would have given us the limit of 1, and we need to apply some other convergence test. But the radius of convergence is only concerned about x values where the absolute value is less than r. And what that means is that for any actual value of x, any e-qualities become inequalities. So we don't really need to worry about these undecided cases, at least not yet. Or let's take another example. Let's find the radius of convergence of 1 over n, x to the n. So while x is raised to the nth power, 1 over n is not, so the ratio test is probably easier. Again, since we want the series to converge, we require the limit of the ratio to be strictly less than 1, which gives us... And so applying our favorite convergence test, we find that convergence requires the absolute value of x be strictly less than 1. And so the radius of convergence is r equal to 1.