Error Approximation: Alternating Power Series

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 you’ve 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: 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?

5. Taylor’s Theorem The difference between a function at x and it’s 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(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.

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”.