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Intro to Analysis in-class_Part_48

# Intro to Analysis in-class_Part_48 - 7.2 Numerical Series...

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7.2 Numerical Series and Sequences 101 Theorem 7.2.13. Suppose we have a vector space ( X, k·k ) . 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 Δ ineq < ε, for m >> 1 , since k =1 k x k k converges, k = N k x k k N →∞ ------→ 0 by Tail-Conv 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 ε > 0 , N, such that m, n N = ⇒ k x n - x m k < ε, or j N , n j , such that m, n n j = ⇒ k x n - x m k < 1 2 j . So we can find a subsequence { x n j } , choosing n 1 < n 2 < . . . . Define y 1 = x n 1 , y j = x n j - x n j - 1 , j > 1 .

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Intro to Analysis in-class_Part_48 - 7.2 Numerical Series...

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