 So, there's not a lot to say about finite series. They have a finite number of terms, and if you want to know what their sum is, add them up. And if the finite series is an arithmetic or geometric series, there is a corresponding sum formula, and if it's not, we can just add the terms. But what happens if our series has an infinite number of terms? To that end, we have to introduce the following idea. An infinite series produces an infinite sequence through what are called the partial sums. Given an infinite series, the k-th partial sum is the sum of the terms up to the k-th term of the series. And one note of caution. We define our partial sums by the term that we end on, so the k-th partial sum ends with the series term ak. Because our series itself might start at a0, or a1, or a2000, the actual number of sum ends is going to vary depending on what series we're looking at. So, for example, let's consider the following. We will find the first four partial sums of the series whose terms are 1 over n, and here we start our series at n equals 1. So our first partial sum, which will designate s sub 1, is the sum up to the n equals 1 term of the series. And that's just one term, which is going to be 1, and so s1 is 1. That's not exactly a world-shaking pronouncement, so let's look at s2. This is the sum of the terms of the series up to n equals 2. So our first term of the series, when n equals 1, is 1 over 1, is 1. And our second term, n equals 2, is going to be 1 half. So s2, our second partial sum, is going to be 1 plus 1 half, or 3 halves. Okay, so now things are starting to get interesting. s3 is going to be the sum up to the n equals 3 term of the series. And so that sum will be... Now, we could just add these three terms together, but we might as well take advantage of the fact that we already know what 1 plus 1 half is. That's the second partial sum, that's s2, and we already know that value, which will simplify the problem of finding s3. S4 will be the sum up to the n equals 4 term of the series, which is going to be... And we already know the sum of the first three terms of the series, which gives us s4. And this leads to our following definition for the convergence of an infinite series. If the partial sums of an infinite series have a limiting value, the series is said to converge to that value. Otherwise, the series diverges. For example, let's take the sum of the series starting with 1 and going on to infinity of minus 1 to power i and determine whether our series converges or diverges. So in order to answer that, we need to look at the partial sums and see if they seem to approach a limiting value. So s1 is the sum of the first one term of the series. s2 is the sum of the series up to the i equals 2 term. s3 is the sum of the series up to i equals 3. s4 is the sum of the series up to i equals 4. And let's take stock of our results. Our partial sums form the sequence negative 1, 0, negative 1, 0, and it seems like these values will continue to alternate between negative 1 and 0. Since the partial sums do not appear to have a limit, then we say that our series is divergent. This leads to a series of problems, but the most obvious is, given a series, find its sum if it exists. So let's try it out. The series, it's an infinite series, will either find the sum if it exists or prove that it does not. In order to determine convergence, we need to know what the partial sums look like. Now you might recognize that this is an arithmetic sequence, and you might remember how to find the sum of an arithmetic sequence. As for myself, I always forget, so let's go ahead and figure this out using the basic idea of how we found that formula. For convenience, we'll index our partial sums by the last term, so that s sub n is going to be our sum up to the n entry of our series. So s sub n is we're going to start with our first term, that's when lowercase n equals 5, that's 17, plus n equals 6 gives us the term 20, and so on, up to the last term, when n equals capital N, that's going to give us 3n plus 2, and it's convenient to include the term immediately before that one, which will be 3n minus 1. You can either calculate that by noting that that occurs when lowercase n equals capital N minus 1, or because this is an arithmetic sequence with a common difference of 3, the term before is going to be 3 less. So remember we summed these arithmetic series by reversing all of our terms, and that gave us a whole bunch of sum n's that were exactly the same, 3n plus 19, which gave us twice the sum of the series itself. We do have to figure out how many terms we have, and so the observation we might make is the first one term of the series is when n equals 5, the last term is when n equals capital N, and we might notice that our term number is 4 less than the value of n, so there's n minus 4 terms in our series. So the sum of this set of 3n plus 19's, there's n minus 4 of them, but if I add them all together, I get twice the sum that I want, so our partial sum should be one half that amount. And that tells us what the partial sums are, and we want to find the limit as n goes to infinity of these partial sums, which turns out to be infinity, so the series diverges. Now the preceding worked because we had a formula for the sum of an arithmetic series, or we could figure out the sum of the arithmetic series even if we didn't remember what the formula was, and we have formulas for the sum of an arithmetic series, a geometric series, or a telescoping series. But what happens if the series isn't arithmetic, geometric, or telescoping? We have a couple of options at this point. We could find a formula for the partial sums, and it's wonderful if we can do that. In fact, that's research-level mathematics. However, that might be too difficult for us to do in most cases because it's research-level mathematics. So instead, we could fall back on a different idea. If the series converges, then the partial sums approximate the series value. And so once I know that the series is actually convergent, what I can do is I can find the partial sum of the first, oh, I don't know, ten billion terms, and this will give me an approximation to the actual value of the series. And so this raises the real problem that we have to answer. Suppose I have an infinite series. Determine whether the series converges.