# hw6 - Homework 6 Math CS 120 Winter 2009 Due on Thursday...

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Homework 6 – Math CS 120, Winter 2009 Due on Thursday, February 19th, 2009 1. Given A R n , and z R n , prove that the following statements are equivalent: (a) z A (the closure of A , also denoted by cl ( A )). (b) There exists a sequence { x n } n N A such that x n z . (c) For any ± > 0, A B ( z , ± ) 6 = {∅} . 2. Prove that for any A R n , ∂A = cl ( A ) \ int ( A ), where cl ( A ) denotes the closure of A , and int ( A ) denotes its interior. 3. Consider a convergent sequence, { x n } ⊂ R p , and its limit, z R p . Deﬁne the set A = { x n } n R p ∪ { z } . Prove, using the deﬁnition in terms of open coverings, that A is compact. 4. Show that 2 1 - p < x p + y p ( x + y ) p < 1 for any x > 0, y > 0, and p > 1. 5. Discuss the convergence behavior of the following series stating in each case whether the series is absolutely convergent, conditionally conver- gent, or divergent: (a) X n =1 1 n ! . (b) X n =3 n ( n + 1)( n + 2) ( n + 3) 3 . (c) X n =1 ± 1 + 1 n 2 ² n 2 . (d) X n =1 ( - 1) n n . 1

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2 (e) X n =1 log ( 1 + 1 n
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hw6 - Homework 6 Math CS 120 Winter 2009 Due on Thursday...

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