SECTION 8 2 SERIES SERIES v If we

SECTION 8. 2 SERIES

SERIES v. If we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a 3 + ··· + an + ··· 8. 2 P 2

INFINITE SERIES v. This is called an infinite series (or just a series). n It is denoted, for short, by the symbol. v. However, does it make sense to talk about the sum of infinitely many terms? 8. 2 P 3

INFINITE SERIES v. It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ··· + n + ··· n n If we start adding the terms, we get the cumulative sums 1, 3, 6, 10, 15, 21, . . . After the nth term, we get n(n + 1)/2, which becomes very large as n increases. 8. 2 P 4

INFINITE SERIES v. However, if we start to add the terms of the series we get: 8. 2 P 5

INFINITE SERIES v. The table shows that, as we add more and more terms, these partial sums become closer and closer to 1. n In fact, by adding sufficiently many terms of the series, we can make the partial sums as close as we like to 1. 8. 2 P 6

INFINITE SERIES v. So, it seems reasonable to say that the sum of this infinite series is 1 and to write: v. We use a similar idea to determine whether or not a general series (Series 1) has a sum. 8. 2 P 7

INFINITE SERIES v. We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 + a 4 n In general, 8. 2 P 8

INFINITE SERIES v. These partial sums form a new sequence {sn}, which may or may not have a limit. v. If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σan. 8. 2 P 9

Definition 2 v. Given a series its nth partial sum: , let sn denote v. If the sequence {sn} is convergent and exists as a real number, then the series Σan is called convergent and we write: n n The number s is called the sum of the series. Otherwise, if the sequence {sn} us divergent, then the series is called divergent. 8. 2 P 10

SUM OF INFINITE SERIES v. Thus, the sum of a series is the limit of the sequence of partial sums. n So, when we write , we mean that, by adding sufficiently many terms of the series, we can get as close as we like to the number s. v. Notice that: 8. 2 P 11

SUM OF INFINITE SERIES VS. IMPROPER INTEGRALS v. Compare with the improper integral n n To find this integral, we integrate from 1 to t and then let t → ∞. For a series, we sum from 1 to n and then let n → ∞. 8. 2 P 12

Example 1 v. An important example of an infinite series is the geometric series v. Each term is obtained from the preceding one by multiplying it by the common ratio r. n We have already considered the special case where a = ½ and r = ½ earlier in the section. 8. 2 P 13

GEOMETRIC SERIES v. If r = 1, then sn = a +… + a = na → ±∞ n Since doesn’t exist, the geometric series diverges in this case. v. If r ≠ 1, we have sn = a + ar 2 + … + ar n– 1 and rsn = ar + ar 2 + … +ar n– 1 + ar n 8. 2 P 14

GEOMETRIC SERIES v. Subtracting these equations, we get: sn – rsn = a – ar n v. If – 1 < r < 1, we know from Result 8 in Section 8. 1 that r n → 0 as n → ∞. 8. 2 P 15

GEOMETRIC SERIES v. So, n Thus, when |r | < 1, the series is convergent and its sum is a/(1 – r). v. If r ≤ – 1 or r > 1, the sequence {r n} is divergent by Result 8 in Section 8. 1 v. So, by Equation 3, does not exist. n Hence, the series diverges in those cases. 8. 2 P 16

GEOMETRIC SERIES v. Figure 1 provides a geometric demonstration of the result in Example 1. v. If s is the sum of the series, then, by similar triangles, v. So, 8. 2 P 17

GEOMETRIC SERIES v. We summarize the results of Example 1 as follows. The geometric series is convergent if |r | < 1 and the sum of the series is: If |r | ≥ 1, the series is divergent. 8. 2 P 18