Section 9.6 Infinite Series: “Alternating Series; Absolute and Conditional Convergence”
power series & radius of convergence
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Transcript of power series & radius of convergence
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Taylor and Maclaurin Series
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Taylor and Maclaurin Series
![Page 4: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/4.jpg)
Taylor and Maclaurin Series
![Page 5: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/5.jpg)
Taylor and Maclaurin Series
![Page 6: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/6.jpg)
Taylor and Maclaurin Series
![Page 7: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/7.jpg)
Taylor and Maclaurin Series
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Example 1
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Example 1 – Solution
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Taylor and Maclaurin Series
![Page 11: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/11.jpg)
Taylor and Maclaurin Series
![Page 12: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/12.jpg)
Taylor and Maclaurin Series
![Page 13: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/13.jpg)
Taylor and Maclaurin Series
![Page 14: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/14.jpg)
Taylor and Maclaurin Series
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Taylor and Maclaurin Series
![Page 16: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/16.jpg)
Taylor and Maclaurin Series
![Page 17: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/17.jpg)
Taylor and Maclaurin Series
![Page 18: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/18.jpg)
Taylor and Maclaurin Series
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Taylor and Maclaurin Series
![Page 20: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/20.jpg)
Taylor and Maclaurin Series
![Page 21: power series & radius of convergence](https://reader036.fdocuments.us/reader036/viewer/2022062503/587789d51a28abc85f8b70a9/html5/thumbnails/21.jpg)
Example 2
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Example 2 – Solution
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Taylor and Maclaurin Series
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Example 8
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Example 8 – Solution
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Example 8 – Solution
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DEFINATION
A power series about x=0 is a series of the form
=+++…….++……… A power series about x=a is a series of the form
=+++….++…. In which the center a and the coefficieants , , ,…., are constants.
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THE CONVERGENCE THEOREM FOR POWER SERIES
If the power series =+++……. Converges for x=c ≠0,then it converges absolutely for all x with |x|<|c|. If the series diverges for x=d , then it diverges for all x with |x|>|d|..
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RADIUS OF CONVERGENCE In previous explainations there is a number R so that power
series will converge for , |x – a|< R and will diverge for |x – a|> R.This number is called the radius of convergence for the series.Note that the series may or may not converge if |x – a| =R .What happens at these points will not change the radius of cnvergencce.
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