11.10: Representations of Functions as Power Series

If you are absent, check out the appropriate AP Classroom videos.

11.10 Notes.

You have already watched 3Blue1Brown, but it is worth putting here again: The Essence of Calculus, Taylor Series! The entire "Essence of Calculus" series on 3Blue1Brown is wonderful. I recommend ALL of them!

Why convergence matters:

Series can be used to approximate functions by using a partial sum of the series
An infinite series = partial sum + the rest of the terms; therefore, (series) - (partial sum) = error
- we need to know where it is a good approximation of the function: whether and where it converges.
- we need to have a way of measuring how much error is left after we stop adding terms

Stuff you must know: all the convergence tests & geometric series rules! Summary of Convergence Tests

Start Memorizing the 5 Common Power Series ! (Note, I don't need you to have the ln series memorized).

Homework: 11.10.1, 11.10.2, & 11.10.3 are due when we return from break... please pace yourself as indicated. It is NOT productive to do these all at once. The point is to do a few problems at a time, and do them well, so you really understand them. The point is NOT to do them all at once in a rush because "they're due tomorrow"!

11.10.1: #7, 11, 13-16 all; and 29.

11.10.2: #23-28 all.

11.10.3: #30-36 all.

Resources:

Two of the best resources for review of Taylor Series are the 2019 AP review videos by Bryan Passwater and Tony Record. They are amazing. We have the newer videos now, but these helped the pandemic BC kids pass the test and they can help you, too. Note, they start with quite a bit of introduction, and none of the testing dates/information is accurate, but you can skip ahead to content:

Finding Taylor Polynomial Approximations of Functions, Part 1

Finding Taylor Polynomial Approximations of Functions, Part 2

Series Homework Solutions

AP Videos!!!

10.11: Finding Taylor Polynomial Approximations of Functions

10.14: Finding Taylor or Maclaurin Series for a Function

10.1: Defining Convergent and Divergent Series
10.2: Working with Geometric Series
10.3: nth Term Test for Divergence
10.4: Integral Test for Convergence (2 videos)
10.5: Harmonic Series and p-series (1 video)
10.6: Comparison Tests for Convergence
10.7: Alternating Series Test
10.8: Ratio Test for Convergence
10.9: Determining Absolute or Conditional Convergence
10.10: Alternating Series Error Bound
10.13: Radius and Interval of Convergence of a Power Series
10.15: Representing Functions as Power Series