Error Approximation:Alternating Power Series

•What are the advantages and limitations of graphical comparisons?

•Alternating series are easy to understand.

•Frequently asked on free-response section of AP test.

Now that you’ve found a polynomial to approximate your function, how good is your polynomial?

Find the 4th degree Maclaurin polynomial for x

xxf

sin

For what values of x does this polynomial best follow the curve? Where does the polynomial poorly follow the curve?

What are the limitations of graphically analyzing a Taylor polynomial?

Example Write the 2nd degree Maclaurin polynomial for:

xy 1

Show that this polynomial approximates y(0.5) to better than 1 part in 100.

Error Approximation:Taylor’s Theorem and Lagrange Error Bounds

• How can we get a handle on how well our polynomial approximates the function for non-alternating series?

• Taylor’s Theorem:• What does it say?• Basically, it’s an existence theorem. What

other existence theorem’s have we seen in Calculus?

• Why is our estimation method called the Lagrange Error Bound?

Taylor’s Theorem

The difference between a function at x and it’s nth degree Taylor polynomial centered at a is:

1

1

!1

nn

n axn

cfxR

for some c between x and a.

Example

Write the 3rd degree Taylor polynomial, P(x), for centered at x= 0.

xey 2

Estimate the error in using P(.2) to approximate . 4.0e

Example

What happens to the Lagrange error bound for the nth degree Maclaurin polynomial for y = sin(x) as n becomes larger and larger?

What does this prove?

Interval of Convergence:Using Geometric Series

•Begin new concept by relating to previous knowledge.•Opportunity to review/teach geometric series if necessary.•Not only learning to find interval of convergence of a series, but also learning why!•Learning new concepts and reviewing old concepts concurrently.

Example

Find the interval of convergence for the following power series: ...421 2 xxxf

Example

Using the formula for geometric series, find the power series for the following function:

291

1

xxf

For what values of x does this power series converge?

What does this mean?

Interval of Convergence:The Ratio Test

•The Ratio Test is the workhorse of all of the tests.•Answers the question: After sufficiently many terms, does this series behave like a geometric series?•Teach in the context of convergence intervals.•For finding intervals, other tests generally needed only at endpoints.•What are intervals of convergence for cos(x), etc.?

Example

For what values of x does the following power series converge?

0

2

23n

n

nx

nxg

Example

What is the interval of convergence for the

Maclaurin series for ? 2cos xy

Series Convergence:Harmonic Series and Alternating Series

•Example: y = ln(1+x)•For what values of x is the ratio test useless?•Does the Harmonic series converge? Integral test.•Does the Alternating Harmonic series converge? Alternating series test (which has already been discussed!) and absolute vs. conditional convergence.•Practice both with convergence of particular series and with intervals of convergence for power series.

Series Convergence:Fun with series

•Geometric Series•Alternating Series Test•Integral Test•“P” test•Comparison Tests•Telescoping Series (a chance to review partial fractions).

Interval of Convergence Flow Chart

Series Convergence Flow Chart

Advantages

•Makes more sense.

•Most important concepts introduced early.

•Series convergence motivated by need to understand intervals on which Taylor series is valid.

•Can get through chapter faster.

Disadvantages

•More work initially for you.

•Less reliance on textbook for you and your students.

•“Non-traditional”.