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Unformatted text preview: Math 301/ENAS 513: Homework 5 Solution Triet Le October 14, 2007 1. Let K be a fixed natural number and let be a map from N to N such that is 1to1 from { ( n 1) K, ( n 1) K + 1 , ..., nK } onto itself, for all n = 1 , 2 , 3 ... . Suppose n =0 a n converges. Show that n =0 a ( n ) converges to the same limit as n =0 a n . Find a convergent series and a rearrangement such that n =0 a ( n ) diverges. Solution 1 . Let s m = n n =0 a n and t m = m n =0 a ( n ) . Since maps { , ..., nK } onto itself. we have s nK = t nK . Now pick any m , there exists an n s.t. nK &lt; m ( n +1) K . We have t m = t nK + a ( nK +1) + ... + a ( m ) = s nK + a ( nK +1) + ... + a ( m ) . Since s m s and a m 0. Thus, for each &gt; 0. There exists an N s.t.  s m s  &lt; and  a m  &lt; , for all m N . This implies, for m N ,  t m s   s nK s  + m X j = nK +1  a ( j )  &lt; + K = ( K + 1) ....
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This note was uploaded on 07/18/2008 for the course MATH 301 taught by Professor Trietle during the Fall '07 term at Yale.
 Fall '07
 TrietLe
 Math

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