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This sum is telescopic all terms but the first and

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This sum is telescopic. All terms but the first and the last cancel, hence s n = ln(1) - ln( n + 1) = - ln( n + 1). As n → ∞ , ln( n + 1) → ∞ , and the sequence of partial sums { s n } is divergent. 38. (10 points) Express the number 6 . 2 54 as a ratio of integers. Answer: 6 . 2 54 = 344 55 Solution: Remark that 6 . 2 54 = 6 . 2 + 54 10 3 + 54 10 5 + . . . . Therefore the number 0 . 0 54 is a sum of the geometric series with a = 54 10 3 and r = 1 100 , so 0 . 0 54 = a 1 - r = ( 54 10 3 ) 1 - 1 100 = ( 54 10 3 ) ( 99 100 ) = 54 990 = 3 55 . We conclude that 6 . 2 54 = 62 10 + 3 55 = 31 5 + 3 55 = 341 55 + 3 55 = 344 55 . 42. (10 points) Find the values of x for which the series n =0 ( x +3) n 2 n converges. Find the sum of the series for those values of x . Answer: - 5 < x < - 1 , S = - 2 x +1 Solutions: The series is a geometric series with a = 1 and r = x +3 2 , it converges if x + 3 2 < 1 ⇔ | x + 3 | < 2 ⇔ - 5 < x < - 1 . The sum equals to a 1 - r = 1 1 - x +3 2 = - 2 x + 1 . Section 8.3 12.(10 points) X n =1 ( n - 1 . 4 + 3 n - 1 . 2 ) . 2
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Answer: Convergent. Solution: n =1 n - 1 . 4 and n =1 n - 1 . 2 are two convergent p -series with p = 1 . 4 > 1 and p = 1 . 2 > 1 respectively. Therefore their sum is convergent as well. 19. (10 points) Determine whether the series n =1 cos 2 n 1+ n 2 converges or diverges.
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