8'8 Power Series

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8.8 Power Series
Mrs. Kessler
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From the text
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A power series is in this form
or
The coefficients a0, a1, a2 are constants.
The center c is also a constant.
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p
That was easy lets learn more.
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Through the Taylor Polynomial process we learned
to represent ex by the following
For what values of x will this series on the
right converge to ex ?
The answer to that is what this section is all
about.
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There are three possibilities for power series
convergence.
The series converges over some finite
interval (the interval of convergence).
The series may or may not converge at the
endpoints of the interval. Discussion for
tomorrow.
The number R is the radius of convergence.
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The number R is the radius of convergence.

• The set of all values of x for which the power
  series converges is called the
• Interval of Convergence.

How do we find the radius of convergence?
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Consider the Ratio Test.
0
Therefore for any fixed value of x, the limit
equals 0. So by the Ratio Test, the series
converges for all x. The radius of convergence R
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which is its Center.
converges only at x 0
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Consider the Ratio Test.
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Ex
Use the Ratio Test.
FYI what it looks like.
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Ex
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Ex
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Ex
Note If R is infinite, then the series converges
for all values of x.
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Find the radius of convergence
R 1 -1 lt x lt 1
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Endpoint Convergence
What happens at the endpoints?
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Case 1.
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Case 2.
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Try this one.
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HW

• p 623 5 - 37 eoo find interval of convergence
• which you should have already done without the
  endpoints, now also find interval including
  endpoints.