Representing Functions by Power Series
A power series
is said to represent a function f with a domain equal to the interval I of convergence of the series if the series converges to f(x) on that interval.
That’s if:
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Example
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Examples
Example(1)
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Solution
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Example(2)
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Solution
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Question
What about the convergence at the end points?
1. The function ln(x-1) is not defined at x = 1
2. We can show easily that the series is convergent if x = -1 (how?)
But does it converge to ln2?
The answer to this question has to
wait till we introduce Able’s Theorem
Approximating ln2
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Example(3)
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Solution
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Question
What about the convergence at the end points?
We can show easily that the series is convergent if x = 1or x = -1 (how?)But does it converge to arctan1 = π/4 & arctan(-1) = π/4 respectively ?
The answer to this question has to
wait until after we introduce Able’s Theorem!
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Question
Approximate 3√e
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Home Quiz (2)
Homework
2
4
3
2
2
4
)1()()()9(
)arctan()()8(
3arctan)()7(91
1)()6(
)23ln()()5(
)3(
1)()()4(
3)()3(
23
5)()2(
3
1)()1(
,
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