 So far, all the series we've looked at have terms that are specific real numbers. We can generalize this concept into what's known as a power series. And these are very important and useful types of series. A power series in x is an infinite series of the form, sum of terms that look like an times x to power n. Now, if we write down the first few terms of such a series, we see that our terms look like a0 plus a1x plus a2x squared plus a3x cubed and so on. And we see that a power series appears to be a polynomial. Actually, it isn't because a polynomial has a last term and because this is an infinite series, there is no last term of the series. So the power series itself is not a polynomial. However, what makes the power series useful is that the partial sums are polynomials and if the series converges, the partial sums approximate the series itself and so we have a polynomial approximating an infinite sum. And there's the big problem. Depending on the value of x, the series might converge or diverge. And so we'll introduce the following definition. The values x for which the power series converges form the interval of convergence. And this leads to a new and important problem. Find the interval of convergence for a given power series. Let's go through our tests. The nth term test tends to give us either inconclusive answers or tells us that the series is divergent. And this isn't what we want. Remember, we want to find the values of x that will make our power series converge and a test that can only tell us that the series diverges or gives us no information isn't useful. How about the integral test? Well, that's a really powerful test and it will tell us whether the series converges or diverges except it's too hard to use in general. And again, not what we want. How about the comparison and limit comparison test? The only problem is they are too specific to be useful for finding all values of x that lead to convergence. We have to be able to compare our power series to a specific series whose convergence or divergence that we know, and that's hard to do. And what this leaves is the ratio and the root test. So let's take a look at those. Say we want to find the interval of convergence for the series 1 over n x to the n. Let's apply the ratio test. So we'll take a look at the limit as n goes to infinity of the ratio between two successive terms. At n, we have a term 1 over n x to the n and the term after it 1 over n plus 1 x to power n plus 1. And we'll do a little bit of algebraic simplification. Now the first thing to recognize is that our limit is as n goes to infinity which means that x isn't changing and so we could remove it from the limit. However, we need to be a little bit careful. We do have these absolute value and x could be a negative number. So when we remove it to the front of our limit operator that x has to appear as an absolute value of x. Meanwhile, since n is supposed to be a positive whole number n over n plus 1 doesn't need those absolute value operators so we can drop them and then take the limit as n goes to infinity which will just be the absolute value of x. At this point we can play the game of wishful thinking. In life, the game of wishful thinking is not actually useful but in mathematics it's frequently useful because we can often get what we want. What we want is the series to converge. In order for the series to converge the limit that we get from the ratio test must be less than 1. So if the absolute value of x is less than 1 the series automatically converges. Now remember that if the limit is 1 the ratio test is inconclusive. So if the absolute value of x is equal to 1 the ratio test doesn't tell us whether or not the series converges or diverges. Now here we do pay a price for our wishful thinking. On the one hand we found where the series definitely converges. On the other hand we have these inconclusive cases and here's the important thing to remember. In conclusive cases are not allowed. We have to see what happens if absolute value of x is actually equal to 1. So if absolute value of x is equal to 1 then we know that x is either equal to 1 or equal to negative 1. If x is equal to 1 the series becomes which is a harmonic series and we know this is divergent. So x equals 1 has to be excluded from our interval of convergence. If x equals negative 1 our series becomes the alternating harmonic series which is convergent and so we should include x equals negative 1 in our interval of convergence and that gives us our interval of convergence. One more thing. Mathematicians like to record as much information as possible about solutions to problems and interval of convergence is no exception. While the important feature of this problem is finding the interval of convergence, negative 1 less than or equal to x strictly less than 1 an incidental byproduct appears here where we determined that the series will converge if the absolute value of x is strictly less than 1. This amount 1 in some sense tells you how far away from a center you can get and still have convergence and because of that we talk about this value here in this case 1 as the radius of convergence. The radius of convergence will become important in advanced mathematics but for series and sequences in calculus 2 the important thing to focus on is finding this interval of convergence. How about another example? Say we want to find the interval of convergence for this series. Since the terms of the series are powers it's probably easiest to use the root test so we'll apply the root test and find the limit as n goes to infinity of the nth root of the absolute values of the terms. We'll do some algebra. Remember absolute value of x doesn't change as n goes to infinity so we can remove that to the front of our limit operator. We can find the limit and again playing the game of wishful thinking. The root test will give us convergence if our limit is less than 1 so we want 2 times the absolute value of x to be less than 1 so that means the absolute value of x is less than 1 half. So if the absolute value of x is less than 1 half the series automatically converges. And again if we want to make use of it this 1 half is the radius of convergence. Again remember that the root test gives us inconclusive results if the limit is equal to 1 but we're not allowed to leave inconclusive cases and so we need to check what happens at the end points of our interval. So if x equals plus or minus 1 half the root test is inconclusive so we'll check it out. At x equals 1 half our series is and this series because its terms don't go to zero definitely diverges. At x equals negative 1 half the series is and again because the terms of this series don't go to zero this series also diverges. So neither 1 half nor negative 1 half should be part of our interval of convergence and our interval of convergence will just include the interval between minus 1 half and 1 half exclusive of the end points.