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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
|∑ | where is an arbitrary small number. Hence the sum approaches 0 as n→∞. 2...
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This note was uploaded on 04/28/2013 for the course ENEE 222 taught by Professor Simon during the Spring '13 term at Maryland.
- Spring '13