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Unformatted text preview: 7.2 Numerical Series and Sequences 101 Theorem 7.2.13. Suppose we have a vector space ( X, kk ) . Then X is complete iff every absolutely convergent series in X converges. Proof. ( ) Suppose that every Cauchy sequence in X converges and that k =1 k x k k converges. Must show that k =1 x k converges. Show that the sequence of partial sums is Cauchy, hence converges. Let s n = n k =1 x k . Then for n > m , we have k s n s m k = n X k =1 x k m X k =1 x k = n X k = m +1 x k n X k = m +1 k x k k by ineq < , for m >> 1 , since k =1 k x k k converges, k = N k x k k N  0 by TailConv Thm. ( ) Suppose that k =1 k x k k converges = k =1 x k converges . Use this to show that any Cauchy sequence converges. Let { x n } be Cauchy. Then > , N, such that m,n N = k x n x m k < , or j N , n j , such that m,n n j = k x n x m k < 1 2 j ....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.
 Fall '11
 Wong
 Vector Space

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