Test1 Fall 2011

Assume that this is the case and focus on the second

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: only if |a|<1. Assume that this is the case and focus on the second term. Let m be an integer between 0 and n-1. Consider the sum ∑ ∑ ∑ The first term on the right can be written ∑ ∑ . If we take |r|<1, then this term can be bounded above in absolute value by ∑ ∑ ∑ ∑ For any fixed m this term goes to zero as n→∞. Now consider the second term ∑ . Again using the fact |a|<1, we can bound this from above by ∑ . This is a segment of the geometric series ∑ , which we know converges for |r|<1. Therefore, the sums of terms “far out” in the series must be getting small. That means, that given any small positive number , if n is large enough, we can choose an m<n so that the ∑ . And so, we have shown that if n is large enough, we can bound the sum by |∑ | where is an arbitrary small number. Hence the sum approaches 0 as n→∞. 2...
View Full Document

This note was uploaded on 04/28/2013 for the course ENEE 222 taught by Professor Simon during the Spring '13 term at Maryland.

Ask a homework question - tutors are online