Geometric Series Before we define what is meant by a series, we need to introduce a related topic, that of sequences. Formally, a sequence is a function that computes an ordered list. Suppose that on day 1, you have 1 dollar, and every day you double your money. Then the function f(n) = 2n generates the sequence 1, 2, 4, 8, 16, 32, …, when n = 1, 2, 3, 4, 5, 6, … This list represents the amount of dollars you have after n days. Note: The use of “…” is read as “and so on”. The individual entries in a sequence are called the terms of the sequence. In our discussion, we are going to assume that the terms in a particular sequence are real numbers. Sequences can be grouped into two large classes based upon the number of terms they include. An infinite sequence is a function that has the set of natural numbers as its domain. As the name implies, it contains an infinite number of terms. In the opening example, the use of the “…” without some number on the end implies that the sequence continues indefinitely, following the prescribed pattern. Of course, there is an inherent problem with assuming that money can be doubled forever. Instead, it makes sense to talk about doubling money for a certain number of days. Say, for n = 1, 2, 3, 4, 5, 6, and 7. In that case, the sequence generated would be called a finite sequence. Its domain is equal to a finite set of natural numbers. (In this case, D = {1, 2, …, 7}.) A common notation for sequences is let an = f(n). With this notation, we say that an is the nth term in the sequence. Example 1: Write out the first five terms a1, a2, a3, a4 and a5 of the following sequences. (a) (1)n na (b) 2sinnan (c) 25 nan(d) 12(3)nna 1Solution: (a) a1 = (-1)1 = -1, a2 = (-1)2 = 1, a3 = (-1)3 = -1, a4 = (-1)4 = 1, a5 = (-1)5 = -1. (b)  122sin (1) sin 1a, 22sin (2) sin 0a,  3322sin (3) sin 1a, 42sin (4) sin 2 0a,  5522sin (5) sin 1a. (c) , , 12(1) 5 3a 22(2) 5 1a 32(3) 5 1a, , 42(4) 5 3a 52(5) 5 5a . (d) , 11 012(3) 2(3) 2a21 122(3) 2(3) 6a, 31 232(3) 2(3) 18a, 41 342(3) 2(3) 54a, . 51 452(3) 2(3) 162a It is worth noting that using these formulas we would easily compute the 1,000th term in the sequence. We would only need to plug in n = 1000. Some sequences are not written in terms of an explicit function like those above. Instead, they may be defined recursively, and hence are called a recursive sequence. That is, each term after the first few terms are defined in terms of what has come before it. If it happens that the terms in our sequence are multiplies of each other (as was the case in Example 1d), then we say that we have a geometric sequence. In Example 1d, the multiple between each term was 3. We call this number the common ratio and it is usually denoted by an r. A geometric sequence can be defined recursively based on the common ratio between terms. That is, we have the relationship an = ran – 1. If we know the starting term of our sequence, a1, since there is a common ratio r between subsequent terms, we can find an explicit formula for the nth term of the sequence. Let us work out a few terms and try to discover the underlying pattern. a2 = ra1 a3 = ra2 = r(ra1) = r2a1 a4 = ra3 = r(ra2) = r(r(ra1)) = r3a1 In general, we have 2The nth term of a Geometric Sequence In a geometric sequence with first term a1 and common ratio r, the nth term, an, is given by an = a1rn – 1 Example 2: Find a formula for the geometric sequence given by 2, 1, 1/2, 1/4, 1/8, … Solution: The first term is 2, so that is a1. Notice that the common ratio between subsequent terms is 1/2 . So, we have that r = 1/2. Thus, an = 2(1/2)n – 1. Example 3: Find a general term an for the following geometric series if a2 = 4 and a4 = 64. Solution: We know that a2 = a1r and a4 = a1r3. So, if 4 = a1r and 64 = a1r3, we can divide the two equations to get 64/4 = (a1r3)/(a1r) = r2, and we see that r = 4. Plugging that into the first equation, we 4 = a1(4), so a1 = 1. Thus, an = 1(4)n – 1 = 4n – 1. Now that we have established what is meant by a sequence and in particular a geometric series, we can turn our attention to a series. Recall, a sequence is a function that computes an ordered list. A series, on the other hand, is the summation of elements generated by a sequence. Let us return to our example with doubling money that we opened with. That is, suppose that on day 1, you have 1 dollar, and every day you double your money. Let’s change the scenario slightly. Suppose on day 1 you have 1 dollar, but every day you are given twice the amount that you had the previous day. The function that specifies how much money you receive on the nth days is given by f(n) = 2n. This generates the sequence 1, 2, 4, 8, 16, 32, …, when n = 1, 2, 3, 4, 5, 6, … Suppose our interest is how much money you have after n days. (Remember, the above values are only how much you receive on a particular day. 3You still get to keep your money from the previous days!) We would need to sum the values of our sequence up until day n to answer this question. As was the case with sequence, series can be grouped into two large classes based upon the number of terms they include. An infinite series is the summation of the terms in an infinite sequence. A finite series is the summation of the terms in a finite sequence. We shall consider both types of series. We define a geometric series as the summation of the terms in a geometric sequence. We can use the formula for the nth term of the geometric sequence to develop a formula for the sum of the first n terms in a geometric sequence. Recall, if a1 was the first term in the geometric sequence with a common ratio of r, then the formula for the nth term in a geometric sequence is given by an = a1rn – 1. Let Sn denote the sum of the first n terms in a geometric sequence. Then we have: Sn = a1 + a1r + a1r2 + … + a1rn – 2 + a1rn – 1 Multiplying this equation by r, we have rSn = a1r + a1r2 + … + a1rn – 1 + a1rn