Chapter 8 Infinite Series

Convergence of Infinite Series Section 8.1 Convergence of Infinite Series

We begin with some notation. Let (an) be a sequence of real numbers. We use the notation to denote the sum am + am + 1 + … + an, where n  m. Using (an), we can define a new sequence (sn) of partial sums given by We also refer to the sequence (sn) of partial sums as the infinite series If (sn) converges to a real number s, we say that the series is convergent and we write We also refer to s as the sum of the series . A series that is not convergent is divergent. If lim sn = + , we say that the series diverges to + , and we write

As defined above, the symbol is used in two ways: It is used to denote the sequence (sn) of partial sums. It is used to denote the limit of the sequence (sn) of partial sums, provided that this limit exists. The context will make the intended meaning clear. Remember that represents a limit of partial sums. In particular, an expression such as a1 + a2 + a3 + … + an + …, really means…

Example 8.1.1 For the infinite series , we have the partial sums given by This is a “telescoping” series because of the way the terms in the partial sums cancel.  1, as n   So we write or

Example 8.1.2 Theorem 8.1.4  (kan) = ks, for every k  . The harmonic series has the partial sums In Example 4.3.13 we saw that the sequence (sn) is divergent. Thus the harmonic series is also divergent. Since the partial sums form an increasing sequence, we must have lim sn = + , so Theorem 8.1.4 Suppose that  an = s and  bn = t with s, t  . Then  (an + bn) = s + t and  (kan) = ks, for every k  . The proof follows from the corresponding result for sequences.

(Cauchy Criterion for Series) Theorem 8.1.5 If  an is a convergent series, then lim an = 0. Proof : If  an converges, then the sequence (sn) of partial sums must have a finite limit, say s. But an = sn – sn – 1, so lim an = lim sn – lim sn – 1 = s – s = 0.  Note that the converse of Theorem 5 is not true. The harmonic series is divergent, even though lim 1/n = 0. Corresponding to the Cauchy criterion for sequences (Theorem 4.3.12), we have the following result for series. Theorem 8.1.6 (Cauchy Criterion for Series) The infinite series  an converges iff for each  > 0 there exists a natural number N such that, if n  m  N, then | am + am + 1 + … + an | < .

Example 8.1.7 One of the most useful series is the geometric series . In Exercise 3.1.7 we saw that for r  1 we have for all n  . But Exercise 4.1.7(f ) shows that lim r n + 1 = 0 when | r | < 1, so we conclude that for | r | < 1. If | r |  1, then the sequence (r n) does not converge to zero, and it follows from Theorem 8.1.5 that the series  r n diverges. Here is a geometric view of the geometric series:

The Geometric Series Slope is r Slope is 1 Let A = (0,0) and B = (1,0). C Draw a line of slope r through A, with 0 < r < 1. Draw a line of slope 1 through B. Let C be the point where they meet. Let P1 be the point on AC directly above B. How long is BP1? P1 Slope is r r A 1 B Slope is 1

The Geometric Series Slope is r Slope is 1 Let P2 be the point on BC at the same height as P1. C How long is P1 P2? Let P3 be the point on AC directly above P2. P5 r3 How long is P2 P3? r3 Repeat this process. P3 r2 P4 r2 P1 r P2 Slope is r r A 1 B Slope is 1

The Geometric Series . . . s0 s1 s2 s3 s4 s Now project the horizontal lengths down onto the x-axis. C And project down the point C as well, and call it D. Look at the partial sums: P5 s0 = 1 s1 = 1 + r r3 s2 = 1 + r + r2 s3 = 1 + r + r2 + r3 r3 P3 r2 P4 etc. r2 And P1 r P2 r A 1 B r r2 r3 r4 D . . . s0 s1 s2 s3 s4 s

The Geometric Series s0 s1 s2 s3 s4 s . . . How long is CD? A B 1 C r 1 s0 s1 s2 s3 s4 C r r2 r3 P1 P3 P5 s . . . D P2 P4 It’s the same as BD. Look at the similar triangles ABP1 and ADC. The corresponding sides are proportional. s 1 s – 1 r = s – 1 r4

The Geometric Series . . . s0 s1 s2 s3 s4 s Solve this for s. s 1 = C sr = s – 1 P5 s – sr = 1 s(1 – r) = 1 r3 P3 r2 P4 s = 1 1 – r s – 1 r2 P1 r P2 r A 1 B r r2 r3 r4 D . . . s0 s1 s2 s3 s4 s

The Alternating Geometric Series Slope is – r Slope is 1 A B 1 r2 P3 P4 r r3 r2 P6 P5 r3 P2 r P1

The Alternating Geometric Series Slope is – r Slope is 1 1 A B r2 P3 P4 r4 r r3 r5 r 4 r2 P6 P5 r3 P2 r P1

The Alternating Geometric Series Slope is – r Slope is 1 s0 s1 s3 s5 s s4 s2 1 A D B r2 P3 P4 Look at the partial sums: r4 r s0 = 1 r3 r5 r 4 s1 = 1 – r C r2 s2 = 1 – r + r2 P6 P5 r3 P2 r P1

The Alternating Geometric Series Slope is – r Slope is 1 s0 s1 s3 s5 s s4 s2 1 A D B P3 P4 Similar triangles give us r r3 s 1 1 – s r = r5 r 4 C r2 or 1 1 + r s = P6 P5 r3 P2 r P1

Practice 8.1.8 A geometric series with r = 2/3. Find the sum of the series . We have = 2(3) = 6. And, what is ? This series is the same as the first series, except without the first term. So,