121616949-math.251 - 1-1 x 1 = 1-0 = 1 Thus the sequence...

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10.1 Sequences 237 Likewise the Squeeze Theorem ( 4.1 ) becomes THEOREM 10.3 Suppose that a n b n c n for all n > N , for some N . If lim n →∞ a n = lim n →∞ c n = L , then lim n →∞ b n = L . And a final useful fact: THEOREM 10.4 If lim n →∞ | a n | = 0 then lim n →∞ a n = 0. This says simply that if the size of a n gets close to zero then in fact a n gets close to zero. Here are some alternate notations for sequences that are frequently used: a 0 , a 1 , a 2 , a 3 , . . . = { a i | i = 0 , 1 , 2 , 3 . . . } = { a i | i Z 0 } = { a i } i =0 EXAMPLE 10.5 Determine whether n n + 1 n =0 converges or diverges. If it con- verges, compute the limit. Since this makes sense for real numbers we consider lim x →∞ x x + 1 = lim x →∞
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Unformatted text preview: 1-1 x + 1 = 1-0 = 1 . Thus the sequence converges to 1. EXAMPLE 10.6 Determine whether ‰ ln n n ± ∞ n =1 converges or diverges. If it converges, compute the limit. We compute lim x →∞ ln x x = lim x →∞ 1 /x 1 = 0 , using L’Hˆopital’s Rule. Thus the sequence converges to 0. EXAMPLE 10.7 Determine whether { (-1) n } ∞ n =0 converges or diverges. If it converges, compute the limit. This does not make sense for all real exponents, but the sequence is easy to understand: it is 1 ,-1 , 1 ,-1 , 1 . . ....
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