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Unformatted text preview: then A B R n + p is compact . 7. Let U R n be open, and consider a function f : U R m . Prove that f is continuous if and only if A R m open set, f-1 ( A ) R n is open. 8. Determine whether the following series converges or diverges: X n =2 1 n (log n ) 2 . 1 2 9. Prove that if lim n x n = x, then lim 1-(1- ) X n =1 x n n = x. Prove that the converse is, in general, false by computing the second limit for x n = (-1) n . 10. Consider the series of real numbers X n =1 a n . Prove that if lim sup n a n +1 a n = r < 1 , the series converges, and that if r > 1, the series diverges. Find two examples, one converging and one diverging, for which r = 1....
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This document was uploaded on 12/27/2011.
- Fall '09