010 100 points Determine whether the following series s n 1 p 3 n 2 3 5 n 2 P n

# 010 100 points determine whether the following series

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010 10.0 points Determine whether the following series s n =1 p 3 - n 2 3 + 5 n 2 P n is absolutely convergent, conditionally con- vergent, or divergent. 1. conditionally convergent 2. divergent 3. absolutely convergent correct Explanation: The given series has the form s n =1 a n where a n = p 3 - n 2 3 + 5 n 2 P n . But then | a n | 1 /n = v v v v 3 - n 2 3 + 5 n 2 v v v v . Thus lim n →∞ | a n | 1 /n = 1 5 < 1 . Consequently, by the Root Test, the given series is absolutely convergent . 011 10.0 points Determine whether the following series 4 5 + 4 · 7 5 · 7 + 4 · 7 · 10 5 · 7 · 9 + 4 · 7 · 10 · 13 5 · 7 · 9 · 11 + ··· is absolutely convergent, conditionally con- vergent, or divergent. 1. divergent correct 2. absolutely convergent
choice (hac762) – HW Quest Week 8 – cepparo – (53850) 6 3. conditionally convergent Explanation: One way to do this problem is to observe that the numerators of the sequence which is being summed are given recursively by p 1 = 4 , p n +1 = p n (4 + 3 n ) , while the denominators are given by q 1 = 5 , q n +1 = q n (5 + 2 n ) . The sequence we are interested in summing is a n = p n q n , which by these recursion relations can be writ- ten a n +1 = p n +1 q n +1 = p n (4 + 3 n ) q n (5 + 2 n ) = a n p 4 + 3 n 5 + 2 n P . Thus lim n →∞ v v v v a n +1 a n v v v v = lim n →∞ 4 + 3 n 5 + 2 n = 3 2 > 1 , so by the Ratio Test the original series is divergent . 012 10.0 points Determine whether the series s n =1 2 · 4 · 6 · . . . · (2 n ) n ! is absolutely convergent, conditionally con- vergent, or divergent. 1. absolutely convergent 2. conditionally convergent 3. divergent correct Explanation: s n =1 2 · 4 · 6 · . . . · (2 n ) n ! = s n =1 (2 · 1) · (2 · 2) · (2 · 3) · . . . · (2 n ) n ! = s n =1 2 n n ! n ! = s n =1 2 n It diverges by the divergence test. 013 10.0 points Which, if either, of the following statements are true? A. The Ratio Test can be used to deter- mine whether the series s n = 1 1 n 2 converges or diverges. B. The Root Test can be used to determine whether the series s k = 1 p ln( k ) 2 + k P k converges or diverges. 1. A only 2. B only correct 3. both of them 4. neither of them Explanation: A. False: when a n = 1 /n 2 , then v v v v a n +1 a n v v v v = p n n + 1 P 2 -→ 1 as n → ∞ , so the Ratio Test is inconclusive.
choice (hac762) – HW Quest Week 8 – cepparo – (53850) 7 B. True: when a k = p ln( k ) 2 + k P k , then | a k | 1 /k = ln( k ) 2 + k -→ 0 as k → ∞ , so a k is convergent by the Root Test. 014 10.0 points If lim n →∞ a n = 0, which, if any, of the following statements are true: (A) lim n →∞ ( a n ) 2 = 0 , (B) s n a n is convergent . 1. neither A nor B 2. A only correct 3. B only 4. both A and B Explanation: (A) TRUE: by Properties of Limits, lim n →∞ ( a n ) 2 = ± lim n →∞ a n ² 2 = 0 , (B) FALSE: when a n = 1 /n , then lim n →∞ a n = 0, but s n = 1 a n = s n =1 1 n diverges by the Integral Test. 015 10.0 points Determine which, if any, of the series A.