# 3: Series

• • Contributed by Marcia Levitus

Chapter Objectives

• Learn how to obtain Maclaurin and Taylor expansions of different functions.
• Learn how to express infinite sums using the summation operator ($$\displaystyle \Sigma$$)
• Understand how a series expansion can be used in the physical sciences to obtain an approximation that is valid in a particular regime (e.g. low concentration of solute, low pressure of a gas, small oscillations of a pendulum, etc).
• Understand how a series expansion can be used to prove a mathematical relationship.

• 3.1: Maclaurin Series
A function f(x) can be expressed as a series in powers of x as long as f(x) and all its derivatives are finite at x=0.
• 3.2: Linear Approximations
We can always approximate a function as a line as long as x is small. When we say ‘any function’ we of course imply that the function and all its derivatives need to be finite at x=0 .
• 3.3: Taylor Series
Before discussing more applications of Maclaurin series, let’s expand our discussion to the more general case where we expand a function around values different from zero. Let’s say that we want to expand a function around the number h. If h=0, we call the series a Maclaurin series, and if h≠0 we call the series a Taylor series. Because Maclaurin series are a special case of the more general case, we can call all the series Taylor series and omit the distinction.
• 3.4: Other Applications of Mclaurin and Taylor series
• 3.5: Problems

Thumbnail: The graph shows the function $$\displaystyle y=sinx$$ and the Maclaurin polynomials $$\displaystyle p_1,p_3$$ and $$\displaystyle p_5$$. Image used with permission (CC BY-SA 3.0; OpenStax).