mat25-hw9solns

mat25-hw9solns - Math 25: Advanced Calculus UC Davis,...

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UC Davis, Spring 2011 Math 25 — Solutions to Homework Assignment #9 1. Decide whether or not each of the following infinite series converges or diverges. Prove your claims. (a) The series n =1 1 n n diverges. Proof. Consider the sequence a n = n n . Note that lim n →∞ log ( n n ) = lim n →∞ log n n = 0 , which implies a n 1. So, by the n th term test (also called the trivial test in the textbook), the series diverges. (b) The series n =1 1 n n converges. Proof. Note that n k n j for n,k,j N and k < j , so that 1 n k 1 n j . Then, X n =1 1 n n = 1 + X n =2 1 n n 1 + X n =2 1 n 2 , so the series converges by the comparison test. (c) The series n =1 n ( n +1) ( n +2) 2 diverges. Proof. Note that lim n →∞ n ( n + 1) ( n + 2) 2 = 1 , so the series diverges by the n th term test. (d) The series n =1 3 n ( n +1)( n +2) n 3 n diverges. Proof. Note that X n =1 3 n ( n + 1)( n + 2) n 3 n = X n =1 3 n 3 + 9 n 2 + 6 n n 3 n = X n =1 3 n 3 n 3 n + 9 n 2 n 3 n + 6 n n 3 n 3 X n =1 1 n , which diverges by the p -series test. So, by the comparison test, the original series diverges. 1
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mat25-hw9solns - Math 25: Advanced Calculus UC Davis,...

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