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?