Intro to Analysis in-class_Part_48

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 > , 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 ....
View Full Document

This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online