 We'll find it useful to be able to re-index a series, so let's take a close look at that. First, we'll introduce the usual disclaimer, provided we're at a point where the series converge absolutely, we can add, subtract, multiply, integrate, or differentiate power series term-wise, and still get a convergent power series. However, it would be nice to write it as a power series, so we might say that the product of two power series is a power series, but it'd be nice to know what it is. In order to write a sum, difference, product, integral, or derivative of a power series as a power series, we need to re-index. So let's establish a few ground rules. The basic simplification we can make for power series is that the sum of two power series is going to be the series of their sum. Well, that seems pretty easy, except there's a lot of fine print here. Let's zoom in so we can read the fine print. In order to make the simplification, both series must begin at the same value and end at the same value. Here, we're both starting at k and continuing on to infinity. The other requirement is that both series must have the same power on x. So here, both terms have power x to n. And so it's important to keep in mind, to add series term-wise, the series must begin and end at the same place and must have the same power on x. Incidentally, as long as you have the same starting and ending place and the same power on x, the name of the index variable doesn't matter. You can switch it around. So these two series both begin at k and go to infinity. And if I rename the index variable n, I can add them term-wise. And what this usually means is we need to change the expression of the terms. So first, we may need to modify the power on x. We'll say that a power series is in standard form if it's written as a sum from k to infinity of an x to power n. In other words, we want our power on x to be the same as the index variable. So we might want to rewrite as a power series in standard form the sum from 1 to infinity nx to power n minus 1. So we want the power on x to be n. Well, currently it's n minus 1, so we'll define a new variable, k equal to n minus 1. And so that means that n will be k plus 1. Equals means replaceable. So since our series begins at n equal to 1 and k is equal to n minus 1, our series will begin at 0. And so making the substitution k equals n minus 1 makes our exponent k, our coefficient n, k plus 1, and our series goes from k equals 0 to infinity. And finally, since it doesn't matter what we call the index variable, we'll go with tradition and call it n. So our new series is going to be going from n equals 0 to infinity with coefficients n plus 1 x to power n. The other requirement if we want to add to series is that we also need to begin our series at the same point. And for that, we can rely on the following fact. The series from k to infinity is the same as the series from k to anywhere we want, plus the series from the next term onto infinity. In other words, we can split off the first few terms of the series and start wherever we want. So for this series, if we want to rewrite to include a series in standard form starting at n equals 2, we can just split off our series. So we want to include a power series starting at 2. Since our series actually starts at 0, what this means is that we've split off the 0 and 1 terms. And since this is a finite series, we can write out those first two terms explicitly. And so our series can be written this way, where we have a series that begins at 2 and runs onto infinity plus some leftover terms. So let's rewrite this series in standard forms with additional terms as needed. So a useful guideline here is adjust the powers first, then the starting point. So our first series is already in standard form. Our power is the index, so we can leave it alone. The second series needs to be adjusted. So in the second series, we want our power to equal the index. So we'll make the substitution n minus 2 equals k. Now the series as written has an n and an n minus 1. So we'll want to write those in terms of k. So n and n minus 1 are equal to k plus 2 and k plus 1. And our starting point, n equals 2, becomes k equals 0. And so making these substitutions in our second series we get, and again it doesn't matter what we call the index variable, so reduce, reuse, recycle, we'll call it n. Now while the two series are now written in standard form, they are not starting at the same point. This one starts at n equals 2, this one starts at n equals 0. And since our series started different points, we need to adjust them. And the thing that's helpful to keep in mind is we can always start later. And what that means is the series that starts at n equals 0, we can adjust so it starts at n equals 2. But we can't adjust the series that starts at n equals 2 and start it earlier. So we'll adjust our second series to start at n equals 2. And so that means we're going to start at n equals 2, but we'll have the first two terms separate. And to gain street cred among gangs of rogue mathematicians, we can write out what these first two terms of the series are going to be. And so our first two terms can't be included in the summation. But because our two series begin at the same point, end at the same point, and have the same powers on x, we can combine the coefficients.