Understanding Infinite Series

1. SECTION 8.2 SERIES

2. SERIES • If we try to add the terms of an infinite sequence we get an expression of the forma1 + a2 + a3 + ··· + an + ··· 8.2

3. INFINITE SERIES • This is called an infinite series (or just a series). • It is denoted, for short, by the symbol. • However, does it make sense to talk about the sum of infinitely many terms? 8.2

4. INFINITE SERIES • It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ··· + 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

5. INFINITE SERIES • However, if we start to add the terms of the serieswe get: 8.2

6. INFINITE SERIES • The table shows that, as we add more and more terms, these partial sums become closer and closer to 1. • 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

7. INFINITE SERIES • So, it seems reasonable to say that the sum of this infinite series is 1 and to write: • We use a similar idea to determine whether or not a general series (Series 1) has a sum. 8.2

8. INFINITE SERIES • We consider the partial sums s1 = a1 s2 = a1 + a2 s3 = a1 + a2 + a3s3 = a1 + a2 + a3 + a4 • In general, 8.2

9. INFINITE SERIES • These partial sums form a new sequence {sn}, which may or may not have a limit. • If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σan. 8.2

10. Definition 2 • Given a series , let sn denote its nth partial sum: • If the sequence {sn} is convergent and exists as a real number, then the series Σan is called convergent and we write: • 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

11. SUM OF INFINITE SERIES • Thus, the sum of a series is the limit of the sequence of partial sums. • 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. • Notice that: 8.2

12. SUM OF INFINITE SERIES VS. IMPROPER INTEGRALS • Compare with the improper integral • 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

13. Example 1 • An important example of an infinite series is the geometric series • Each term is obtained from the preceding one by multiplying it by the common ratior. • We have already considered the special case where a = ½ and r = ½ earlier in the section. 8.2

14. GEOMETRIC SERIES • If r = 1, then sn = a + a +…+ a = na → ±∞ • Since doesn’t exist, the geometric series diverges in this case. • If r≠ 1, we havesn = a + ar + ar2 + … + ar n–1and rsn = ar + ar2 + … +ar n–1 + ar n 8.2

15. GEOMETRIC SERIES • Subtracting these equations, we get: sn– rsn = a – ar n • If –1 < r < 1, we know from Result 8 in Section 8.1 that r n → 0 as n → ∞. 8.2

16. GEOMETRIC SERIES • So, • Thus, when |r | < 1, the series is convergent and its sum is a/(1 –r). • If r≤ –1 or r > 1, the sequence {r n} is divergent by Result 8 in Section 8.1 • So, by Equation 3, does not exist. • Hence, the series diverges in those cases. 8.2

17. GEOMETRIC SERIES • Figure 1 provides a geometric demonstration of the result in Example 1. • If s is the sum of the series, then, by similar triangles, • So, 8.2

18. GEOMETRIC SERIES • 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