 Welcome back to our lecture series, Math 1220, Calculus II for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. This video represents the first of lecture 36 in our series, which will start section 11.2 about series, which are closely related to the notion of sequences we've talked about before. Now, lecture 36 is mostly going to focus on the idea of a geometric series, very important family of series that we'll talk about very shortly. We first need to introduce what is the idea of a series. So imagine we have a sequence, a sequence will denote it as just a sub n right now. So this is just a list of numbers. Now from a sequence, we could construct a new sequence, which is referred to as the sequence of partial sums. And it's given by the following formula, s sub n is going to equal the sum where k ranges from one to n of the sequence ak. That is to say, if we expand the sum, we're taking a one plus a two plus a three plus a four all the way up to here, a sub n. So if we just take a quick example of such a thing, we could take the sequence a sub n equals one over two to the n. So this is supposed to be here a sequence. And so our sequence would look like one half, one fourth, one eighth, one sixteenth. You get the idea there. And so it's sequence of partial sums. We would take the first term, which is the first term here. We would just add up together one half. That's all there is. The second term, we take one half plus one fourth, which is going to give us three fourths. The third term would be the sum of the previous terms, one half plus a fourth plus an eighth, which this will add up to be seven eighths. The next term would be one half plus one fourth plus one eighth plus one sixteenth, which is going to give us fifteen sixteenths, like so. And so the members of this sequence would be one half, three fourths, seven eighths, sixteen or fifteen sixteenths. And so we create a new sequence from a previous sequence by adding together all the terms. So this is a sequence in its own right, of course. And as because as it's a sequence, this partial sum sequence, we could ask things like is the sequence bounded? Is it monotonic, increasing, decreasing? Is it convergent? Does it have a limit, what seems to be happening here? And although this this information we related to the original sequence, this information could be distinct, could be distinct from the previous sequence. So for example, if we take the sequence a sub n right here, this is a bounded decreasing sequence, which is convergent. It has to be convergent by the monotone convergence theorem. Notice that our sequence a sub n, it'll be bounded between one and zero. And in fact, this sequence is decreasing. It's decreasing. And we have that a sub n will approach will approach zero as n goes to infinity. But on the other hand, if you look at the sequence of partial sums, this sequence s sub n, it will likewise sit between, I realize I wrote it backwards earlier, my mistake, and you should switch this around. It sits between one and zero. So our sequence s n will sit between one and zero. That part's the same. But notice this sequence as we go from one half to three fourths to seven eighths to five sixteenths, this sequence is actually increasing. And as it's a bounded increasing sequence, it's convergent by the monotone convergence theorem. But in this situation, s n is going to converge towards the number one. So it's convergent, but it goes to a different location, not necessarily the same place as the sequence here. And as such, we're interested in the limit of this sequence of partial sums. So we would take the limit as n goes to infinity of s sub n. But as s sub n is the sum of the, of the sequence, you're taking the sum of the ak's as k equals one here. And as k goes from one to n here, this, we're going to abbreviate using the following notation. We're going to take the sum where k equals one to infinity here of a sub k. That should be a k right there. And this right here is going to become an infinite sum. So we add together a one, a two, a three, a four, a five, a six indefinitely. And this gives us what we refer to as a series. A series is an infinite sum. Now it's not exactly the same infinite sum as a Riemann sum, like we talk about with integrals. And we'll make some distinction about this in the future. But we have this infinite sum. It's the sum of all the terms of a sequence. And we call this a series. Now this thing could be convergent. It could be divergent. It's going to depend on properties of the sequence in play here. But I want to kind of mention that this idea of a infinite sum, these series is actually quite natural, right? Believe it or not, these infinite series can, in fact, converge, converge to finite numbers as kind of alluded to in this example. Take us another example, the sequence of b sub n, whose entry, whose nth entry is going to equal the nth decimal, decimal digit of the number pi. You know, so let's see if pi is 3.149, b1 would be 1, b2 would be 4, b3 would be 1, b4 would be 5, you know, keep on going with the idea there. And so then we're going to see that pi, we often write as 3.14159 dot dot dot, right? This can be expressed as an infinite sum, right? Because this is 3 plus one tenth plus four one hundredths plus one one thousandth plus five ten thousandth plus nine one hundred thousandths. Keep on going. But if we recognize the pattern that's in play here, we could rewrite this thing as 3 plus the sum of, as we allow i to go from one to infinity of our sequence bn over 10 to the i power. Because after all, these numbers 1, 4, 1, 5, 9, these are just the first five numbers in the sequence, which we define over here as bn. So when we look at irrational numbers like pi or e, or even rational numbers, we can express every decimal expansion as this infinite sum of a sequence over powers of 10. And therefore it really, this idea of an infinite series is actually quite natural to our number system. Irrational numbers very well lead to these series. And so we'll see over the next several lectures how important that these infinite series turn out to be.