Copyright © Cengage Learning. All rights reserved. Sequences and Series

2 Copyright © Cengage Learning. All rights reserved Geometric Sequences

3 Objectives ► Geometric Sequences ► Partial Sums of Geometric Sequences ► What Is an Infinite Series? ► Infinite Geometric Series

4 Geometric Sequences

5 An arithmetic sequence is generated when we repeatedly add a number d to an initial term a. However, what happens when we repeatedly multiply a number r to an initial term a ? 3, 6, 12, 24, 48,... (multiplying by 2) A geometric sequence is generated when we start with a number a and repeatedly multiply by a fixed nonzero constant r The number r is called the common ratio because the ratio of any two consecutive terms of the sequence is r.

6 the Nth term of a Geometric Sequence

7 Geometric Sequences A geometric sequence is generated when we start with a number a and repeatedly multiply by a fixed nonzero constant r.

8 Example 1 – Geometric Sequences (b) The sequence 2, –10, 50, –250, 1250,... is a geometric sequence with a = 2 and r = –5. When r is negative, the terms of the sequence alternate in sign. The nth term is a n = 2(–5) n – 1. cont’d

9 Example 1 – Geometric Sequences (c) Given the sequence Is it Geometric? if so find r a = 1 and r = The nth term is. cont’d

10 Finding Terms of a Geometric Sequence We can find the nth term of a geometric sequence if we know any two terms, as the following example shows. Find the eighth term of the geometric sequence 5, 15, 45,.... Solution: To find a formula for the nth term of this sequence, we need to find a and r. Clearly, a = 5. To find r, we find the ratio of any two consecutive terms. For instance, r = = 3.

11 Example 2 – Finding Terms of a Geometric Sequence Thus a n = 5(3) n – 1 The eighth term is a 8 = 5(3) 8 – 1 = 5(3) 7 = 10,935.

12 Partial Sums of Geometric Sequences

13 Partial Sums of Geometric Sequences For the geometric sequence a, ar, ar 2, ar 3, ar 4,..., ar n – 1,..., the nth partial sum is S n = = a + ar + ar 2 + ar 3 + ar 4 + · · · + ar n – 1

14 Partial Sum of Geometric Sequence To find a formula for S n, we multiply S n by r and subtract from S n. S n = a + ar + ar 2 + ar 3 + ar 4 + · · · + ar n – 1 rS n = ar + ar 2 + ar 3 + ar 4 + · · · + ar n – 1 + ar n S n – rS n = a – ar n Now factoring out S n on the left and a on the right S n (1 – r) = a(1 – r n ) So S n = (r ≠ 1)

15 Partial Sums of Geometric Sequences We summarize this result.

16 Example 4 – Finding a Partial Sum of a Geometric Sequence Find the sum of the first five terms of the geometric sequence 1, 0.7, 0.49, 0.343,... Solution: The required sum is the sum of the first five terms of a geometric sequence with a = 1 and r = 0.7. Using the formula for S n with n = 5, we get Thus the sum of the first five terms of this sequence is

17 What Is an Infinite Series?

18 What Is an Infinite Series? An expression of the form = a 1 + a 2 + a 3 + a is called an infinite series. The dots mean that we are to continue the addition indefinitely. As n gets larger and larger, we are adding more and more of the terms of this series. Intuitively, as n gets larger, S n gets closer to the sum of the infinite series, if it exists.

19 Infinite Series

20 What Is an Infinite Series? Now notice that as n gets large, the term of the series, 1/2 n gets closer and closer to 0. So if Then S n gets close to 1. We can write S n  1 as n  In general, if S n gets close to a finite number S as n gets large, we say that the infinite series converges (or is convergent). The number S is called the sum of the infinite series. If an infinite series does not converge, we say that the series diverges (or is divergent). ONLY CONVERGING INFINITE SERIES WILL HAVE SUMS!!

21 Infinite Geometric Series An infinite geometric series is a series of the form a + ar + ar 2 + ar 3 + ar ar n – It can be shown that if | r | < 1, then r n gets close to 0 as n gets large. Therefore, we say that if | r | < 1, the infinite geometric series will converge.

22 Converging Series

23 Infinite Geometric Series Recall the nth partial sum of such a series is given by the formula It follows that S n gets close to a/(1 – r ) as n gets large, or S n  as n  Thus the sum of this infinite geometric series is

24 Sum of an Infinite Geometric Series

25 Example 6 – Infinite Series Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (a) (b) Solution: (a) This is an infinite geometric series with a = 2 and r =. Since | r | = | | < 1, the series converges. By the formula for the sum of an infinite geometric series we have

26 Example 6 – Solution (b) This is an infinite geometric series with a = 1 and r =. Since | r | = | | > 1, the series diverges. cont’d