121616949-math.254 - n 1 n = ± n n 1 n< 1(Again it...

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240 Chapter 10 Sequences and Series EXAMPLE 10.14 Show that { n 1 /n } converges. We first show that this sequence is decreasing, that is, that n 1 /n ( n +1) 1 / ( n +1) . Consider the real function f ( x ) = x 1 /x when x 1. We can compute the derivative, f ( x ) = x 1 /x (1 - ln x ) /x 2 , and note that when x 3 this is negative. Since the function has negative slope, n 1 /n ( n + 1) 1 / ( n +1) when n 3. Since all terms of the sequence are positive, the sequence is decreasing and bounded when n 3, and so the sequence converges. (As it happens, we can compute the limit in this case, but we know it converges even without knowing the limit; see exercise 1 .) EXAMPLE 10.15 Show that { n ! /n n } converges. Again we show that the sequence is decreasing, and since each term is positive the sequence converges. We can’t take the derivative this time, as x ! doesn’t make sense for x real. But we note that if a n +1 /a n 1 then a n +1 a n , which is what we want to know. So we look at a n +1 /a n : a n +1 a n = ( n + 1)! ( n + 1) n +1 n n n ! = ( n + 1)! n ! n n ( n + 1) n +1 = n + 1
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Unformatted text preview: n + 1 ¶ n = ± n n + 1 ¶ n < 1 . (Again it is possible to compute the limit; see exercise 2 .) Exercises 1. Compute lim x →∞ x 1 /x . ⇒ 2. Use the squeeze theorem to show that lim n →∞ n ! n n = 0. 3. Determine whether { √ n + 47-√ n } ∞ n =0 converges or diverges. If it converges, compute the limit. ⇒ 4. Determine whether n 2 + 1 ( n + 1) 2 ff ∞ n =0 converges or diverges. If it converges, compute the limit. ⇒ 5. Determine whether n + 47 √ n 2 + 3 n ff ∞ n =1 converges or diverges. If it converges, compute the limit. ⇒ 6. Determine whether 2 n n ! ff ∞ n =0 converges or diverges. ⇒ 10.2 Series While much more can be said about sequences, we now turn to our principal interest, series. Recall that a series, roughly speaking, is the sum of a sequence: if { a n | n ∈ Z ≥ }...
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