mat25-hw8solns

mat25-hw8solns - Math 25: Advanced Calculus UC Davis,...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 – Solutions to Homework Assignment #8 1. (a) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is {- 1 , 1 } . (b) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is {- 1 , - 2 - 1 2 , 0 , 2 - 1 2 , 1 } . (c) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is [0 , 1]. (d) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = e ; the set of subsequential limits is { e } . (e) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = 1; the set of subsequential limits is { 1 } . (f) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = ; the set of subsequential limits is . 2. (a) For any m N , since { a m +1 ,a m +2 ,... } ⊆ { a m ,a m +1 ,a m +2 ,... } , b m +1 = sup { a m +1 ,a m +2 ,... } ≤ sup { a m ,a m +1 ,a m +2 ,... } = b m . Thus, (
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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mat25-hw8solns - Math 25: Advanced Calculus UC Davis,...

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