Math 301 Problem Set 5 Solutions

Math 301 Problem Set 5 Solutions - Math 301/ENAS 513:...

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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 1-to-1 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 < 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 > 0. There exists an N s.t. | s m- s | < and | a m | < , for all m N . This implies, for m N , | t m- s | | s nK- s | + m X j = nK +1 | a ( j ) | < + K = ( K + 1) ....
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Math 301 Problem Set 5 Solutions - Math 301/ENAS 513:...

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