mat25-hw9 - Math 25 Advanced Calculus UC Davis Spring 2011...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Homework Assignment #9 Homework due: Tuesday 5/31/11 at beginning of discussion section Reading material. Read section 3.4, 3.5, 3.6.1, 3.6.2, 3.6.12 in the text- book. Problems 1. Decide whether or not each of the following infinite series converges or diverges. Prove your claims. (a) X n =1 1 n n (d) X n =1 3 n ( n + 1)( n + 2) n 3 n (b) X n =1 1 n n (e) X n =1 n + 2 ( n 2 + 1) n (c) X n =1 n ( n + 1) ( n + 2) 2 (f) X n =2 ( - 1) n ln( n ) 2. Find the value of the ( hint: telescoping) infinite sum X n =1 1 n ( n + 1)( n + 2) = 1 1 · 2 · 3 + 1 2 · 3 · 4 + 1 3 · 4 · 5 + 1 4 · 5 · 6 + . . . 3. If ( a n ) n =1 ( b n ) n =1 are sequences such that a n , b n 0 for all n 1, and the sums n =1 a n and n =1 b n are convergent, prove that n =1 a n b n converges as well. 4. In this problem we study the convergence of the series n =1 a n , where a n = ( n !) 2 (2 n )! = 1 ( 2 n n ) . (a) Find a simple formula for a n +1 a n , and show that lim n →∞ a n +1 a n = 1 4 .
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