9.9 Power Series

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9.9 Power Series
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A power series is in this form
or
The coefficients c0, c1, c2 are constants.
The center a is also a constant.
(The first series would be centered at the origin
if you graphed it. The second series would be
shifted left or right. a is the new center.)
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Once we have a series that we know, we can find a
new series by doing the same thing to the left
and right hand sides of the equation.
This is a geometric series where r-x.
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Example 4
Given
find
We differentiated term by term.
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Example 5
Given
find
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Example 5
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This series would allow us to calculate a
transcendental function to as much accuracy as we
like using only pencil and paper!
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Example 5
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Example 5
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Example 3
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Example 3
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Example 2
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Example 2
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• Series that we should know

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We have saved the best for last!
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An amazing use for infinite series
Substitute xi for x.
Factor out the i terms.
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This is the series for cosine.
This is the series for sine.
This amazing identity contains the five most
famous numbers in mathematics, and shows that
they are interrelated.
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p. 676 5 - 25 odd

• When a mathematician writes a Fantasy book, will
  the page numbers be imaginary numbers?