11.2: Series & Convergence (the Calculus Edition)

Action Items:

1. 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 beyond what is contained in my summary notes, refer to Stewart 11.2. We will spend at least three days developing and becoming comfortable with these ideas before moving on.

Key Concepts & Terms:

series
partial sums
convergent series
divergent series
geometric series
the Harmonic Series
Test for Divergence
Sums, differences, and multiples of convergent series

General Strategies for Convergence Testing (so far):

1. Apply the divergence test (remember, if $lim_{n ⟶ \infty} a_{n} = 0$ you haven't proven convergence, you've merely disproven divergence!)

2. Look for a geometric pattern (esp. if both numerator and denominator are a power of n)

3. Look for a combination of "familiars"

4. Use Partial Fractions if appropriate and check for telescoping

5. Write out a few terms to look for patterns you might have missed

Why any of this 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 to infinity); therefore, (infinite series) - (partial sum) = (error)
- we need to know where it is a good approximation of the function: this means we need to know whether and where it converges.
- we need to have a way of measuring how much error is left after we stop adding terms.

2. Practice: 11.2.1: p. 742 #1, 2; 3-29 odd; 30-34; 41-45. This is a long assignment, but we have a 3-day weekend you can use to procrastinate.

Things you MUST KNOW:

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:

10.1: Defining Convergent and Divergent Series
10.2: Working with Geometric Series
10.3: nth Term Test for Divergence