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


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.


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


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

Patrick JMT Just Math Tutorials:
Taylor and Maclaurin Series - Example 2
Taylor / Maclaurin Series for Sin (x)
Taylor’s Inequality
Maclaurin/Taylor Series: Approximate a Definite Integral to a Desired Accuracy
Binomial Series - Example 1
Binomial Series - Example 2

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.