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:


Recall Infinite Sums:

In General:


In General:

Three possible outcomes of infinite sums:

or

•Diverges

•Converges

•Neither


In General:

Special Type of Infinite Sum: Power Series

Which gives


In General:


Depends on x


In General:


Depending on x, can either diverge, converge, or neither

ExampleFirst 10 Terms

ExampleFirst 100

Terms

ExampleEventual

ly

Determining Interval of Convergence

Ratio Test

Converges if

Diverges if

If

, test is inconclusive.


Now Note If:

Then


Ratio Test

Converges if

Diverges if

If



Ratio Test

Converges if

Diverges if

If


Is Known as“Radius of Convergenc

e”


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 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?

Review of Power Series and Taylor Polynomials. Infinite Sums and Power Series Recall Infinite Sums:

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