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Calculus III

Course: Calculus III
Topic: Power Series
Subtopic: Taylor, Maclaurin, and Binomial Series

Overview

We have formed power series centered at x=0, but in this lesson we form the Taylor Series for f entered at a value "a". A Taylor series centered at zero is called a Maclaurin Series.

We also introduce Binomial Series and use them to represent polynomials of degree p that are expansions of binomials to the degree p such as (x+1)p where p is a positive integer. The binomial coefficients used in binomial series have ties to Pascal's Triangle, algebra, statistics, probability, and combinatorics.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 12.3.1 Use the Taylor Series formula to form a Taylor polynomial for a function centered at a given "a" value
• 12.3.2 Form a Maclaurin Series using the Taylor Series formula centered at zero
• 12.3.3 Give the interval of convergence of a Maclaurin Series
• 12.3.4 Understand the notation for binomial coefficients and how to expand them
• 12.3.5 Know some of the connections that binomial coefficients have to algebra (eg. binomial expansions), Pascal's Triangle, statistics, probability, and/or combinatorics
• 12.3.6 Know that (1+x)p is a polynomial of degree p and be able to write it as a Binomial Series
• 12.3.7 Understand and be able to apply the Binomial Series Theorem and formula
• 12.3.8 Give the interval of convergence of a Binomial Series
• 12.3.9 Use the binomial series to approximate roots
• 12.3.10 Be able to write a variety of basic functions as Taylor, Maclaurin, or Binomial series
• 12.3.11 Understand and be able to apply the Convergence of Taylor Series Theorem and the Rn(x) remainder formula to find the remainder in a Maclaurin Series for a given function
• 12.3.12 Have handy and be able to use a table containing a variety of functions written as Maclaurin Series to approximate functions or combine functions

Terminology

Terms you should be able to define: Maclaurin Series, binomial coefficients (definition, formula, notation), Binomial Series

Formulae to have in your notes: Taylor Series for a function centered at "a", Maclaurin Series for a function centered at zero, binomial coefficients, Binomial Series, remainder for the convergence of a Taylor Series

Supplemental Resources (recommended)

View slides pages 16-29 of Applications of Taylor Polynomials from Portland CC Mathematics Dept.

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Binomial Series

Selwyn Hollis's Video Calculus: Taylor and Maclaurin Series

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down middle column to "Infinite Series" for related titles.