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Estimating the Sum of a Series
Suppose
is a convergent sequence of positive integers. Let
which is the error if you approximate
using the first
terms. From Theorem
6.3.2
we get the following.
Proposition
6
.
3
.
8
(Remainder Bound)
Suppose
is a continuous, positive, decreasing function on
and
is convergent. Then
Proof
. In Theorem
6.3.2
set
. That gives
But
since
is decreasing and
.
Example
6
.
3
.
9
Estimate
using the first
terms of the series. We have
The proposition above with
tells us that
In fact,
and we hvae
so the integral error bound was really good in this case.
Example
6
.
3
.
10
Determine if
convergers or diverges. Answer: It converges, since
and
converges.
Next:
Tests for Convergence
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The Integral and Comparison
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The Integral and Comparison
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William Stein 2006-03-15