11.7 Summary of Convergence Testing

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

No new notes! We've learned all the convergences tests you are responsible for knowing. Whew. Time to spend some time synthesizing and revising your knowledge and understanding. 🤓

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

Homework:

11.7: #1-38 odd is the minimum; all is better. This is at least a THREE-day assignment -- expect the unit exam next Tuesday or Wednesday (at the latest). Have some questions ready daily, but I won't look for the assignment to be complete until the day of the test.

The ratio test, as stated before, is used extensively to establish the radius and interval of convergence for power series. You need to be good at it. (The root test doesn't show up as much, if that helps; but Absolute Convergence has been tested recently in FRQs!)

Stuff you must know: ALL the convergence tests & geometric series rules!

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:

AP Videos!!!

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