11.6: Absolute Convergence, and the Ratio and Root Tests for Convergence

Action Items:

1. Absolute Convergence, and the Ratio and Root Tests for Convergence Notes: if you are absent, take notes on the material below as necessary then attempt the homework. If you are struggling with the topic or need a review of concepts, refer to Stewart 11.6 and the AP videos.

BC 11.6 Summary Notes

BC 11.6 Notes with Examples

Why Convergence & Error matter:

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

2. HW: 11.6: #3 - 18 all; 19 - 27 odd; if you finish early, it wouldn't hurt to try evens! Two-day assignment! This does NOT mean don't do anything one day, and try to cram the others into the second day. It means take your time and do them thoroughly and thoughtfully.

The ratio test, in particular, is used extensively to establish the radius and interval of convergence for power series. It is the basis for earning at least two points in most series FRQs. So yeah, you need to be good at this, too. (The root test doesn't show up as much, if that helps; but Absolute Convergence has been tested recently in FRQs!

Things you MUST KNOW:

Ratio Test

Root Test Absolute Convergence

Alternating Series Test

AST Error Bound

Divergence Test

Integral Test

Direct Comparison Test

Limit Comparison Test

P-series Test

Harmonic Series

What kind of series this is:

$LaTeX: a_n=a_1\cdot r^{\left(n-1\right)}$

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

Resources:

Series Homework Solutions

AP Classroom Videos:

Today's Topics:

10.8: Ratio Test for Convergence

10.9: Determining Absolute or Conditional Convergence

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 (2 videos)
10.7: Alternating Series Test
10.10: Alternating Series Error Bound