mat25-hw8 - Math 25: Advanced Calculus UC Davis, Spring...

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Unformatted text preview: Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 Homework Assignment #8 Homework due: Tuesday 5/24/11 at beginning of discussion section Reading material. Read sections 2.11, 2.12, 2.13 in the textbook. Problems 1. For each of the following sequences ( a n ) n =1 , find limsup n a n , liminf n a n , and the set of subsequential limits of ( a n ). (a) a n = (- 1) n (b) a n = sin ( 2 n 8 ) (c) ( a n ) n =1 = { , 1 2 , 1 , , 1 4 , 2 4 , 3 4 , 1 , , 1 8 , 2 8 , 3 8 ,..., 7 8 , 1 , , 1 16 ,..., 15 16 , 1 , , 1 32 ,... } (d) a n = ( 1 + 1 n ) n (e) a n = ( 1 + 1 n 2 ) n Hint: ( 1 + 1 n 2 ) n = h ( 1 + 1 n 2 ) n 2 i 1 /n (f) a n = ( 1 + 1 n ) n 2 Hint: ( 1 + 1 n ) n 2 = ( 1 + 1 n ) n n 2. Let ( a n ) n =1 be a sequence. Denote L = limsup n a n . Define a sequence ( b m ) m =1 by b m = sup { a m ,a m +1 ,a m +2 ,... } . (a) Prove that b m is a nonincreasing sequence....
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mat25-hw8 - Math 25: Advanced Calculus UC Davis, Spring...

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