11.5.2: Alternating Series Test

Today's Topic: Alternating Series Error Bound If you are struggling with the topic or need a review of concepts, refer to Stewart 11.5. Better yet, watch the AP Classroom videos!

Summary of my notes: Error = Infinite - Partial

Remainder (Error) = Actual (Infinite) Sum - Estimate (Partial Sum) and is $\leq$ [at most; i.e. less than or equal to] the Absolute value of the Next Term.

We sometimes say the error is "captured" in "next unused term." The general idea should make sense: each term in a convergent alternating series must be decreasing in size, and alternating in sign. So if you add a bit, then subtract a bit less. . . the greatest amount of error your partial sum can possibly have is the value of the next term. You can see this from the way the graphs of the partial sums oscillate around the value where they are tending (if the graph were connected instead of discrete, it would look like a damped function oscillating around a horizontal asymptote).

This is the first (and simplest) of only two error approximations you are expected to have mastered for the AP test!

A reminder. . . why Convergence & Error 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

Action Items:

1. Notes: if you are absent, take notes on the material (above and) below as necessary then attempt the homework. If you are struggling with the topic or need a review of concepts beyond what is contained in my summary notes, refer to Stewart 11.5. This test is used often, and the AS Error is only one of two approximations for error tested on the exam. Be sure you do everything you can to be comfortable with these!

BC 11.5: Alternating Series Test Notes and/or Alternating Series Summary Notes

BC 11.5: AST Error Bound

2. Practice: 11.5.2: #7-19 all; PLUS #21-29.

Things you MUST KNOW:

Divergence Test

Integral Test

Direct Comparison Test

Limit Comparison Test

Alternating Series Test

Alternating Series Error Bound

p-series Test

Harmonic Series

What kind of series this is: $LaTeX: a_n=a_1\cdot r^{\left(n-1\right)}$

$LaTeX: S=\frac{a_1}{1-r}$ , when LaTeX: |r|<1

Resources:

Series Homework Solutions

AP Classroom Videos:

10.7: Alternating Series Test

10.10: Alternating Series Error Bound

Or catch up with one of the previous AP videos (I highly recommend these, whether you think you fully understand everything or not!)!

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