Error Approximation:

1
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.

2
Now that youve found a polynomial to approximate
your function, how good is your polynomial?
Find the 4th degree Maclaurin polynomial for
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?
3
Example
Write the 2nd degree Maclaurin polynomial for
Show that this polynomial approximates y(0.5) to
better than 1 part in 100.
4
Error Approximation Taylors Theorem and
Lagrange Error Bounds

• How can we get a handle on how well our
  polynomial approximates the function for
  non-alternating series?
• Taylors Theorem
• What does it say?
• Basically, its an existence theorem. What other
  existence theorems have we seen in Calculus?
• Why is our estimation method called the Lagrange
  Error Bound?

5
Taylors Theorem
The difference between a function at x and its
nth degree Taylor polynomial centered at a is
for some c between x and a.
6
Example
Write the 3rd degree Taylor polynomial, P(x), for
centered at x 0.
Estimate the error in using P(.2) to approximate
.
7
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?
8
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.

9
Example
Find the interval of convergence for the
following power series
10
Example
Using the formula for geometric series, find the
power series for the following function
For what values of x does this power series
converge?
What does this mean?
11
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.?

12
Example
For what values of x does the following power
series converge?
13
Example
What is the interval of convergence for the
Maclaurin series for ?
14
Series Convergence Harmonic Series and
Alternating Series

• Example y ln(1x)
• 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.

15
Series Convergence Fun with series

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

16
Interval of Convergence Flow Chart
17
Series Convergence Flow Chart
18
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.

19
Disadvantages

• More work initially for you.
• Less reliance on textbook for you and your
  students.
• Non-traditional.