review of power series and taylor polynomials. infinite sums and power series recall infinite sums:
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Infinite Sums and Power Series Recall Infinite Sums:TRANSCRIPT
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Review of Power Series
and Taylor Polynomials
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Infinite Sums and Power Series
Recall Infinite Sums:
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Infinite Sums and Power Series
Recall Infinite Sums:
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Infinite Sums and Power Series
Recall Infinite Sums:
In General:
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Infinite Sums and Power Series
In General:
Three possible outcomes of infinite sums:
or
•Diverges
•Converges
•Neither
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Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Which gives
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Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depends on x
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Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depends on x
![Page 9: Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:](https://reader034.vdocuments.us/reader034/viewer/2022051010/5a4d1b8e7f8b9ab0599c0198/html5/thumbnails/9.jpg)
Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depending on x, can either diverge, converge, or neither
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ExampleFirst 10 Terms
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ExampleFirst 20 Terms
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ExampleFirst 50 Terms
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ExampleFirst 100
Terms
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ExampleEventual
ly
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Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
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Determining Interval of Convergence
Now Note If:
Then
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Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
![Page 18: Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:](https://reader034.vdocuments.us/reader034/viewer/2022051010/5a4d1b8e7f8b9ab0599c0198/html5/thumbnails/18.jpg)
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
![Page 19: Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:](https://reader034.vdocuments.us/reader034/viewer/2022051010/5a4d1b8e7f8b9ab0599c0198/html5/thumbnails/19.jpg)
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
![Page 20: Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:](https://reader034.vdocuments.us/reader034/viewer/2022051010/5a4d1b8e7f8b9ab0599c0198/html5/thumbnails/20.jpg)
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Is Known as“Radius of Convergenc
e”
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Determining Interval of Convergence
Interval of Convergence
Radius of Convergence
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Power Series Manipulation
Sum:
Derivative:
Reindexing:
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Taylor SeriesFor a Power
Series
It is always true that
So given a function, we can write it as a power series.A power series that describes a function is
called a “Taylor Series”
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Taylor SeriesFor a Power
Series
For Example
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Taylor Series
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Taylor Series
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Taylor Series
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Taylor Series
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Taylor PolynomialsIf we kept going, we would
findno error!
KEY IDEA: We can use Taylor Series (or Power Series)
as substitutes for common functions, because
they ARE THE SAME THING
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Questions?