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Unformatted text preview: Homework #4 Math 131A Due: Feb. 11 Spring 2010 Prof. Maruskin Homework solutions by: Liem Tran. 12.2 Prove that limsup  s n  = 0 if and only if lim s n = 0. Assume limsup  s n  = 0 for n > N , then lim s n = 0 because the absolute values of the supremums reach a limit of 0 which would mean that the sequence s n approached 0 as n , greater than N , increases to infinity. Assume lim s n = 0. For n > N , the absolute value of the limit would approach 0 and as n increases past N then the limit of the supremums would also come to 0. 14.1 Determine which of the following series converges. Justify your answer. (a) ∑ n 4 2 n . a n = n 4 2 n a n +1 a n ⇒ ( n +1) 4 2 n +1 n 4 2 n = ( n + 1) 4 2 n +1 · 2 n n 4 = ( n + 1) 4 2 n 4 = n 4 + 4 n 3 + 6 n 2 + 4 n + 1 2 n 4 = 1 + 4 n + 6 n 2 + 4 n 3 + 1 n 4 2 ⇒ lim 1 + 4 n + 6 n 2 + 4 n 3 + 1 n 4 2 = 1 2 lim a n +1 a n exists so this series converges....
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 '10
 Maruskin,Jared
 Math, Derivative, lim, Limit of a sequence, Limit superior and limit inferior, subsequence

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