By Building Research Establishment Staff, E. Grant, Building Research Establishment
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If ͕a n ͖ is increasing and a n ഛ M for all n, then the terms are forced to crowd together and approach some number L. M L 0 1 23 n 7 Monotonic Sequence Theorem Every bounded, monotonic sequence is FIGURE 10 convergent. EXAMPLE 12 Investigate the sequence ͕a n ͖ defined by the recurrence relation a1 2 for n 1, 2, 3, . . a nϩ1 12 ͑a n ϩ 6͒ SOLUTION We begin by computing the first several terms: Mathematical induction is often used in dealing with recursive sequences. See page 87 for a discussion of the Principle of Mathematical Induction.
4. List the first nine terms of the sequence ͕cos͑n͞3͖͒. Does Find a formula for the general term a n of the sequence, assuming that the pattern of the first few terms continues. 6. 7. ͕2, 7, 12, 17, . ͖ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 9–28 ■ Determine whether the sequence converges or diverges. If it converges, find the limit. 9. a n 3 ϩ 5n 2 n ϩ n2 2n 11. a n nϩ1 3 13. a n ͭ ͑n ϩ 2͒! n! e n ϩ e Ϫn e 2n Ϫ 1 19. ͕n 2e Ϫn ͖ nϩ1 3n Ϫ 1 sn 12. a n 1 ϩ sn ͑Ϫ1͒ nϪ1n 15. a n 2 n ϩ1 17. 10. a n ͮ 14.
5n ϩ 4 2 SOLUTION lim a n lim nlϱ nlϱ n2 1 1 lim 2 2 n l ϱ 5n ϩ 4 5 ϩ 4͞n 5 0 So the series diverges by the Test for Divergence. If we find that lim n l ϱ a n 0, we know that a n is divergent. If we find that lim n l ϱ a n 0, we know nothing about the convergence or divergence of a n. Remember the warning in Note 2: If lim n l ϱ a n 0, the series a n might converge or it might diverge. 1. 9, we have n lim u n lim nlϱ i n l ϱ i1 ϩ bi ͒ lim nlϱ n lim n ͚b ai ϩ i1 ͪ i i1 n ͚a i n l ϱ i1 ͚ͩ n ͚ ͑a ϩ lim ͚b i n l ϱ i1 lim sn ϩ lim tn s ϩ t nlϱ nlϱ Therefore, ͑a n ϩ bn ͒ is convergent and its sum is ϱ ͚ ͑a n n1 ͚a n n1 ϱ EXAMPLE 9 Find the sum of the series ͚ n1 SOLUTION The series ϱ ϩ bn ͒ s ϩ t ͩ ϩ ϱ ͚b n n1 ͪ 3 1 ϩ n .
5m. Steel-Framed Houses by Building Research Establishment Staff, E. Grant, Building Research Establishment