Pre-Calc Homework Solutions 390

Pre-Calc Homework Solutions 390 - 390 Section 9.5 46. (a)...

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42. This is a geometric series which converges only for ) ln x ) , 1, or } 1 e } , x , e . (a) 1 } 1 e } , e 2 (b) 1 } 1 e } , e 2 (c) None 43. n 5 13 3 10 9 ? 365 ? 24 ? 3600 5 4.09968 3 10 17 ln ( n 1 1) , sum , 1 1 ln n ln (4.09968 3 10 17 1 1) , sum , 1 1 ln (4.09968 3 10 17 ) 40.5548. .. , sum , 41.5548. .. 40.554 , sum , 41.555 44. Comparing areas in the figures, we have for all n $ 1, E n 1 1 1 f ( x ) dx , a 1 1 1 a n , a 1 1 E n 1 f ( x ) dx . If the integral diverges, it must go to infinity, and the first inequality forces the partial sums of the series to go to infinity as well, so the series is divergent. If the integral converges, then the second inequality puts an upper bound on the partial sums of the series, and since they are a nondecreasing sequence, they must converge to a finite sum for the series. (See the explanation preceding Exercise 42 in Section 9.4.) 45. Comparing areas in the figures, we have for all n $ N , E n 1 1 N f ( x ) dx , a N 1 1 a n , a N 1 E n N f ( x ) dx . If the integral diverges, it must go to infinity, and the first inequality forces the partial sums of the series to go to infinity as well, so the series is divergent. If the integral
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