Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:
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Transcript of Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:
Review of Power Series
and Taylor Polynomials
Infinite Sums and Power Series
Recall Infinite Sums:
Infinite Sums and Power Series
Recall Infinite Sums:
Infinite Sums and Power Series
Recall Infinite Sums:
In General:
Infinite Sums and Power Series
In General:
Three possible outcomes of infinite sums:
or
•Diverges
•Converges
•Neither
Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Which gives
Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depends on x
Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depends on x
Infinite Sums and Power Series
In General:
Special Type of Infinite Sum: Power Series
Depending on x, can either diverge, converge, or neither
ExampleFirst 10 Terms
ExampleFirst 20 Terms
ExampleFirst 50 Terms
ExampleFirst 100
Terms
ExampleEventual
ly
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of Convergence
Now Note If:
Then
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of Convergence
Ratio Test
Converges if
Diverges if
If
, test is inconclusive.
Is Known as“Radius of Convergenc
e”
Determining Interval of Convergence
Interval of Convergence
Radius of Convergence
Power Series Manipulation
Sum:
Derivative:
Reindexing:
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”
Taylor SeriesFor a Power
Series
For Example
Taylor Series
Taylor Series
Taylor Series
Taylor Series
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
Questions?