hw4 - Math508 Fall 2010 Jerry L Kazdan Problem Set 4 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 4 D UE : Thurs. Oct 7, 2010. Late papers will be accepted until 1:00 PM Friday . 1. a) Calculate lim n 5 n + 17 n + 2 . b) Let a n : = 3 n 2 - 2 n + 17 n 2 + 21 n + 2 . Calculate lim n a n . 2. Investigate the convergence or divergence of a n if a). a n = n + 1 - n b). a n = n + 1 - n n c). a n = 1 1 + z n ( complex z ) 3. Let { a n } and { b n } be any real bounded sequences. a) Show that limsup n ( a n + b n ) limsup n a n + limsup n b n provided the sum on the right is not of the form - . b) Give an explicit example where strict inequality can occur. 4. [Hoffman, p. 36 #10] Let S be a (linear) subspace of R n . If X R n , let P ( X ) be the orthogonal projection of X into the subspace S . If X k converges to X , show that P ( X k ) converges to P ( X ) . 5. If { b k } is a sequence of positive numbers, prove the arithmetic-geometric mean inequality [ b 1 b 2 ··· b n ] 1 / n b 1 + ··· + b n n . When does equality hold?
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