Alternating Series

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Alternating Series
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Alternating Series
Definition
A series of the type
where all the entries ak are positive, is an
alternating series.
Theorem
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Example
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Alternating Series
Theorem
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Proof
The assumptions imply that the partial sums
S2m1 a1-a2a2m-1-a2ma2m1 form a
decreasing sequence.
This follows since S2m1 - S2(m1)1 a2m2
a2m3 0 for all m.
In the same way one sees that S2m
a1-a2a2m-1-a2m form an increasing sequence.
Both sequences are bounded and hence converge.
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Alternating Series
Theorem
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Proof (contd)
It remains to show that
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Alternating Series
Remark
The above estimate implies the following. Assume
that S a1-a2a3-a4 is a converging
alternating series satisfying the conditions of
the previous theorem. The error made when
approximating the sum of the series S by a
partial sum is at most the absolute value of the
first term left out.
Error Estimate
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Alternating Series
Error Estimate
Example
We will see later that S ln(2) 0.6931471806.
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Absolutely Converging Series
Definition
Examples
by the Integral Test.
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Convergence of Absolute Values
Theorem
Proof
Clearly 0 bk ak ak 2ak.
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Convergence of Absolute Values
Theorem
Proof (contd)
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Convergence of Absolute Values
Remark
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Example