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.

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. HW: 11.2.1: p. 742 #1, 2; 3-29 odd; 30-34; 41-45. This is a long assignment; pace yourself to be finished by Friday (two days to complete the work). I will answer questions tomorrow if you have them.

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

Divergence Test

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