121616949-math.252 - -1< r< 0 then | r n | = | r | n...

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238 Chapter 10 Sequences and Series and clearly diverges. EXAMPLE 10.8 Determine whether { ( - 1 / 2) n } n =0 converges or diverges. If it con- verges, compute the limit. We consider the sequence {| ( - 1 / 2) n |} = { (1 / 2) n } . Then lim x →∞ 1 2 x = lim x →∞ 1 2 x = 0 , so by theorem 10.4 the sequence converges to 0. EXAMPLE 10.9 Determine whether { (sin n ) / n } n =1 converges or diverges. If it con- verges, compute the limit. Since | sin n | ≤ 1, 0 ≤ | sin n/ n | ≤ 1 / n and we can use the- orem 10.3 with a n = 0 and c n = 1 / n . Since lim n →∞ a n = lim n →∞ c n = 0, lim n →∞ sin n/ n = 0 and the sequence converges to 0. EXAMPLE 10.10 A particularly common and useful sequence is { r n } n =0 , for various values of r . Some are quite easy to understand: If r = 1 the sequence converges to 1 since every term is 1, and likewise if r = 0 the sequence converges to 0. If r = - 1 this is the sequence of example 10.7 and diverges. If r > 1 or r < - 1 the terms r n get large without limit, so the sequence diverges. If 0 < r < 1 then the sequence converges to 0. If
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Unformatted text preview: -1 < r < 0 then | r n | = | r | n and 0 < | r | < 1, so the sequence converges. In summary, { r n } converges precisely when-1 < r ≤ 1 in which case lim n →∞ r n = n if-1 < r < 1 1 if r = 1 Sometimes we will not be able to determine the limit of a sequence, but we need to know whether it converges. In some cases we can determine this without being able to compute the limit. A sequence is called increasing or sometimes strictly increasing if a i < a i +1 for all i . It is called non-decreasing or sometimes (unfortunately) increasing if a i ≤ a i +1 for all i . Similarly a sequence is decreasing if a i > a i +1 for all i and non-increasing if a i ≥ a i +1 for all i . If a sequence has any of these properties it is called monotonic . EXAMPLE 10.11 The sequence ‰ 2 i-1 2 i ² ∞ i =1 = 1 2 , 3 4 , 7 8 , 15 16 , . . . ,...
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