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
sin x
f x  
x
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:
y  1 x
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 nonalternating 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:
f n1 c 
n 1
x  a 
Rn x  
n  1!
for some c between x and a.
Example
Write the 3rd degree Taylor polynomial, P(x), for y  e
centered at x= 0.
Estimate the error in using P(.2) to approximate
2x
e 0 .4 .
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:
f x   1  2 x  4 x  ...
2
Example
Using the formula for geometric series, find the power
series for the following function:
1
f x  
2
1 9x
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?

2
n
n
g x    n x  2
n 0 3
Example
What is the interval of convergence for the
Maclaurin series for
 
y  cos x 2 ?
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”.