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extracredit - s n → a Conversely give an example such...

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Extra Credit problems, MATH 554/703I 1. Let ( a n ) be a sequence of real numbers such that for some α 6 = 0 we have lim n →∞ a n - α a n + α = 0 . What can you say about ( a n ). 2. Let α, β > 0. Prove that lim n →∞ n p α n + β n = max { α, β } . 3. Assume a n a in R . Put s n = a 1 + ··· + a n n . Prove that also
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Unformatted text preview: s n → a . Conversely give an example such that ( s n ) converges for some divergent sequence ( a n ). 4. Let f : [0 , ∞ ) → R such that f ( x ) = √ x . Prove that f is uniformly continuous....
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