# Is an odd integer because for any odd integer m the

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will converge if and only if m is not an even integer. But for each fixed m , the series X n = 1 m 17 · n is a geometric series with common ratio r = m 17 . This series will converge if and only if - 1 < m 17 < 1, i.e. , if and only if - 17 < m < 17. Thus, for m > 1, the two series will converge if and only if m is an odd integer less than 17. Consequently, m = 3 , 5 , . . . , 15 . keywords: 015 (part 1 of 1) 10 points Determine which, if any, of the following series diverge. ( A ) X n = 1 (4 n ) n n !
Cheung, Anthony – Homework 12 – Due: Nov 21 2006, 3:00 am – Inst: David Benzvi 10 7. C only 8. none of them Explanation: To check for divergence we shall use either the Ratio test or the Root test which means computing one or other of lim n → ∞ fl fl fl a n +1 a n fl fl fl , lim n → ∞ | a n | 1 /n for each of the given series. ( A ) The ratio test is the better one to use because fl fl fl a n +1 a n fl fl fl = 4 n ! ( n + 1)! ( n + 1) n +1 n n . Now n ! ( n + 1)! = 1 n + 1 , while ( n + 1) n +1 n n = ( n + 1) n + 1 n n . Thus fl fl fl a n +1 a n fl fl fl = 4 n + 1 n n -→ 4 e > 1 as n → ∞ , so series ( A ) diverges. ( B ) The root test is the better one to apply because | a n | 1 /n = 3 n + 3 -→ 0 as n → ∞ , so series ( B ) converges. ( C ) Again the root test is the better one to apply because of the n th powers. For then | a n | 1 /n = 3 2 4 n 3 n + 5 -→ 2 > 1 as n → ∞ , so series (C) diverges. Consequently, of the given infinite series, only A and C diverge. keywords: 016 (part 1 of 1) 10 points Which, if any, of the following statements are true? A. If 0 a n b n and X b n diverges, then X a n diverges B. The Ratio Test can be used to determine whether X 1 /n ! converges. C. If X a n converges, then lim n → ∞ a n = 0. 1. A only 2. B only 3. B and C only correct 4. A and B only 5. all of them 6. none of them 7. C only 8. A and C only Explanation: A. False: set a n = 1 n 2 , b n = 1 n . Then 0 a n b n , but the Integral Test shows that X a n converges while X b n diverges. B. True: when a n = 1 /n !, then fl fl fl fl a n +1 a n fl fl fl fl = 1 n + 1 -→ 0
Cheung, Anthony – Homework 12 – Due: Nov 21 2006, 3:00 am – Inst: David Benzvi 11 as n , , so X a n is convergent by Ratio Test. C. True. To say that X a n converges is to say that the limit lim n → ∞ s n of its partial sums s n = a 1 + a 2 + . . . + a n converges. But then lim n → ∞ a n = s n - s n - 1 = 0 . keywords: 017 (part 1 of 1) 10 points Determine which, if any, of the series A. 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . B. X m = 3 m + 2 ( m ln m ) 2 are convergent. 1. A only 2. both of them correct 3. B only 4. neither of them Explanation: A. Convergent: given series is a geometric series X n = 0 ar n with a = 1 and r = 1 2 < 1. B. Convergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x (ln x ) 2 . keywords: 018 (part 1 of 1) 10 points Decide whether the series X m = 3 m ln m ( m + 7) 3 is convergent or divergent. 1. convergent correct 2. divergent Explanation: Since 0 < m ln m ( m + 7) 3 < m ln m m 3 = ln m m 2 , for m 3, the Comparison Test ensures that the given series converges if the series X m
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