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## Taylor Series Introduction

Section:
10.2
Date:
Monday, October 18, 2010 - 14:00 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

Notes on Taylor Series – October 18, 2010

Definition of a Taylor Series

The Taylor series for centered at is Similarly, the Taylor series for centered at is Examples

Since a Taylor series is the same as a Taylor polynomial, but is taken to infinite degree, we already know many Taylor series. From the last section, we know the following.          where There is a bit of a technicality here. In section 10.1, we wrote between the functions and their Taylor polynomials. Now, we're writing instead. ( means “approximately equal to,” while means “equal to.”) The main point to understand here is detailed in the next portion of these notes.

## Taylor Polynomials

Section:
10.1
Date:
Monday, October 11, 2010 - 14:00 - Friday, October 15, 2010 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

Notes on Taylor Polynomials – October 11 & 15, 2010

Definition of a Taylor Polynomial

The Taylor polynomial of degree approximating near is Similarly, the Taylor polynomial of degree approximating near is   This may not at first seem obvious, but if you have calculated the th Taylor Polynomial, you can find the th Taylor polynomial by adding one more term. That is ## Book Assignment 8

Due Date:
Monday, October 18, 2010 - 16:00

Section 10.1 Problems 28, 31 and 32

## Midterm 2 Solutions

Solutions to Midterm 2 are attached.

## Midterm 2 Review Solutions

Math 2300 – Calculus II – University of Colorado

Fall 2010 – Review for Midterm II

1. Rotating the ellipse about the -axis generates an ellipsoid. Compute its volume.

Solution: In this situation, the and would be numbers. But, in general, we have that        This object is symmetric with respect to the -axis and -axis, meaning that we only need to find one of the limits of integration. Setting in the above equation implies When we create such objects, we never have and so we must have and . So, the -value where the ellipsoid has a zero -value must be . (This makes perfect sense from the definition, if you should know it or look it up.)

So, we consider slices in the ellipsoid revolved around the -axis, where the radius is given as and the width of a Riemann disk is . So, we find that the volume is      ## Midterm 2 Review Quiz

Due Date:
Tuesday, October 12, 2010 - 16:00

Name:

Math 2300 Section 005 – Calculus II – Fall 2010

Quiz – Due Tuesday, October 12, 2010

This quiz is intended to be a review, so it contains more problems than usual. I won't be grading each problem individually, but the quiz will be graded as a whole. Solutions will be posted on Tuesday.

1. Find the sum Hint: Write this as two geometric series.

Section:
9.5
Date:
Friday, October 8, 2010 - 14:00 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

Power Series and Radius of Converges – October 8, 2010

Power Series

Definition: A power series is an infinite series of the form where

• represents the coefficient of the th term.

• is some (fixed) real number.

• varies around , and so we sometimes say that the power series is “centered” at .

## Series Convergence Notes

Math 2300 Section 005 – Calculus II – Fall 2010

Review on Series Convergence – October 6, 2010

## § Summary of Convergence of Series

### § Some General Notes

• Pulling a finite number of terms off from a series will not affect convergence / divergence.

• You can pull series apart like integrals (in terms of sums/difference and multiplication by a constant).

• Series notation is shorthand, and in most cases series can be rewritten in any number of different ways. ## Book Assignment 7

Due Date:
Monday, October 11, 2010 - 16:00

Section 9.3 Problem 50
Section 9.4 Problems 60, 84
Section 9.5 Problems 34, 42

## Limit Comparison Test

Section:
9.4
Date:
Wednesday, October 6, 2010 - 14:00

Math 2300 Section 005 – Calculus II – Fall 2010

9.4 Limit Comparison Test Examples – October 6, 2010

1. Determine if the given series converges or diverges using the limit comparison test. Solution: The dominant terms here are in the numerator and in the denominator. So, we compare the series terms The terms form a convergent geometric series. And, we find that Since we obtain a finite positive number using the limit comparison test, we know that both of the series either diverge or they both converge. We already knew that the series of terms converges, so the series in question must also converge.