Chapter 6 Power Series
References.
- Calculus, Early Transcendentals by Stewart, Chapter 11, Sections 8–11.
Introduction.
The goal of this chapter is to learn how to construct series (infinite sums!) whose values solve various problems of interest. For example, I have claimed that we can evaluate the exponential \(e^x\) for any value x by summing the series
\begin{equation*}
\sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots + \frac{x^n}{n!} + \cdots
\end{equation*}
This is not a single series, but a family of them, one for each value of \(x\text{,}\) with that quantity \(x\) appearing only in natural number powers. Thus it resembles a polynomial, but with an infinite number of terms. Such series are called Power Series (or sometimes “infinite polynomials”).
Chapter 6 Review.
When reviewing this chapter, also look at the end of chapter review material in OpenStax Calculus Volume 2, including Key Terms 11 , Key Equations 12 and Key Concepts 13 .
openstax.org/books/calculus-volume-2/pages/6-introductionopenstax.org/books/calculus-volume-2/pages/6-key-termsopenstax.org/books/calculus-volume-2/pages/6-key-equationsopenstax.org/books/calculus-volume-2/pages/6-key-concepts
