Upload
coleen-jordan
View
243
Download
0
Tags:
Embed Size (px)
Citation preview
Calculus II (MAT 146)Dr. Day Monday, Nov 2, 2015
Proving Series Convergence (Chapter 11)
Monday, Nov 2, 2015 MAT 146
Next Test: Thurs 11/19 & Fri 11/20
MAT 146
Fact or Fiction?
Monday, Nov 2, 2015
MAT 146
What is an Infinite Series?
Monday, Nov 2, 2015
We start with a sequence {an}, n going from 1 to ∞, and define {si} as shown.
The {si} are called partial sums. These partial sums themselves form a sequence.
An infinite series is the summation of an infinite number of terms of the sequence {an}.
MAT 146
Notable Series
Monday, Nov 2, 2015
The harmonic series is the sum of all possible unit fractions.
MAT 146
Notable Series
Monday, Nov 2, 2015
A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge?
MAT 146
Notable Series
Monday, Nov 2, 2015
A telescoping sum can be compressed into just a few terms.
MAT 146
Our first Series Convergence Test:
The nth-Term Testalso called
The Divergence Test
Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146
Infinite Series Convergence
Monday, Nov 2, 2015
Our goal is to determine whether an infinite series converges or diverges. It must do one or the other.
If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges.
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146
Converge or Diverge?
Monday, Nov 2, 2015
MAT 146
Converge or Diverge? (9)
Monday, Nov 2, 2015
MAT 146
Converge or Diverge? (12)
Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146
Absolute Convergenceand
Conditional Convergence
Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146
Why Study
Sequences and Seriesin Calc II?
Taylor Polynomials applet
Infinite Process Yet Finite Outcome . . . How Can That Be?
Transition to Proof
Re-Expression!
Monday, Nov 2, 2015
MAT 146
Polynomial Approximators
Monday, Nov 2, 2015
On of our goals this chapter is to generate polynomial functions that can be used to approximate other functions near particular values of x.
The polynomial we seek is of the following form:
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146Monday, Nov 2, 2015
MAT 146
Polynomial Approximators
Monday, Nov 2, 2015
Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
MAT 146Monday, Nov 2, 2015
Create a 3rd-degree polynomial approximator for
MAT 146
Each statement is FALSE! Identify one
word, symbol, or expression and correct it
to make each statement TRUE.
Monday, Nov 2, 2015