 The truth is that sequences aren't actually that important, except that they lead to the notion of a series. And the two are connected in the following way. A series is an ordered sum. In a finite series, the number of terms is finite, and we write that series using the following notation, which is called sigma notation. And so let's take this apart. This an is this generic term of the series, and n ranges from a lowest value of i to a highest value of k. And so the terms of these series are going to be a i plus a i plus 1 plus a i plus 2, and so on all the way up to a k. And we're adding all of these terms together. And the only difference between a finite series and an infinite series is that in an infinite series, the number of terms is infinite. And we write the upper limit as infinity. Now, while we can write down the series as a sum, the actual sum of the series is the value of that sum provided that it exists. So one thing we commonly do is we omit the indices to refer to a generic series. Since a series is an ordered sum, then the terms of that sum form a sequence. Now, if those terms form an arithmetic sequence, we say that the sum is an arithmetic series. If the terms of our series form a geometric sequence, we say that our sum is a geometric series. And if the terms of our series form an alternating sequence, we say that our sum is a, wait for it, alternating series. For example, let's write the first three terms of the series given as shown. Notice that our index starts at 3, so the first three terms of the series correspond to the terms where n equals 3, 4, and 5. So we're given the formula for the individual terms of the series, 5 times 2 to the power n, so we'll let n equals 3, 4, and 5 to find the terms of the series. And these are the first three terms of our sum, but we're omitting a bunch of them, so that will include the dot, dot, dot. Series are a little bit trickier than sequences, but as long as the terms and the order of the terms remain the same, two series are considered identical. What this means, however, is that the indices might not look alike, even though they do give you the same terms in the same order. And because of this, it's sometimes useful to re-index a series. What does that mean? Well, consider a series like this, where our index starts with n equals 5 and ends with n equals 100. We would like to rewrite this as a series beginning at index m equals 1. And in general, it's always a good idea to verify that we have the same thing. So, first of all, we note that the given series starts at n equals 5. Since we want the re-index series to start at m equals 1, then we have m is equal to n minus 4. So, as with the defted integral, what we want to do is we want to make sure that everything is in terms of the same variable. And since our index variable is now m, we'll want to make sure that everything is in terms of m. So, the given series begins when n equals 5, so that tells us that m is equal to 1. And the given series ends when n equals 100, so m is equal to 96. Finally, since m equals n minus 4, then n equals m plus 4. So, the terms of our series, 3n plus 7, can be rewritten in terms of m, as 3m plus 19. Now, having gone through all of this work, let's make sure that our two series are the same thing, and so they should have the same terms in the same order. So, our series begins with n equals 5, so the first term of the series will be 3n plus 7 when n is equal to 5, our next term will be 3n plus 7 when n is equal to 6, our next term will be 3n plus 7 when n is equal to 7, and let's skip to the last term when n equals 100, that last term will be 307. Meanwhile, our re-index series is going to start with m equals 1, so the first term is going to be when m equals 2, our second term is going to be our third term will be when m equals 3, and our last term will be when m equals 96, and the two series begin and end in the same way, so the two series appear to be the same. So, series behave a lot like definite integrals, which shouldn't be any surprise at all because a definite integral is a sum, and a couple of the important properties here for any series at all, the following properties hold. The sum of constant times term of the series is constant times series for any constant c, and any series at all can be split at any point k in between the beginning and end. These properties are useful if we try to combine two series into a single series, so let's say I want to sum n equals 2 to 100 of the series whose terms are 3n plus 5, and add to that the sum from 0 to 100 of the series whose terms look like n squared. Now, an important idea to remember here is that we can only combine series if they begin and end at the same index value. So, the first thing to notice is that the first series runs from 2 to 102 while the second series runs from 0 to 100. This means that if we look at the terms of the series, they overlap in the interval between 2 and 100. So, we'll want to use our ability to break a series apart into a series that runs from 2 to 100 plus whatever the leftover bits are. So, the first series that runs from 0 to 100 can be broken apart into a portion that runs from 0 to 1 and a portion that runs from 2 to 100. But this first bit is just going to consist of the two-term 0 squared plus 1 squared, that's 1, plus the remaining part of the series. And so, we can replace this series from 0 to 100 of n squared with 1 plus the sum from 2 to 100 of n squared. Now, let's take a look at the second series. Again, we'll split off the part that runs from 2 to 100 and then we'll have some leftover terms. We can keep the first part, as that's part of the overlap, and the second part we will evaluate. If n equals 101, the term of the series is 308, and if n equals 102, the term of the series is 311. So, we can add these two terms together and we have an expression for the other series. So, we can replace that. Finally, because the two series have the same beginning and the same ending, we can combine them into a single series by combining the terms n squared plus 3n plus 5. We can also combine the numbers portion 1 and 619, and that gives us an expression for our summed series.