Power Series

$\displaystyle \sum_{n=0}^{\infty} c_n x^n = c_0+c_1x+c_2x^2+c_3x^3+\ldots$ where $x$ is a variable and each $c_n$ is a cofficient, which is a constant (perhaps different constants, depending upon

$n$ ). For example,

$\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!},\quad\text{ or }\quad\sum_{n=0}^\infty \frac{x^n}{n+1}.$ In the first example above,

$c_n=\frac{1}{n!}$ since our terms are

$c_nx^n=\frac{1}{n!}x^n=\frac{x^n}{n!}$ Notice that since

$x$ is a variable, we could plug in

$x=1$ , say, and get the series

$\sum \frac{1}{n!}$ or plug in

$x=\frac{1}{2}$ and get the series

$\sum \frac{1}{2^n n!}$ . DO: What is

$c_n$ for the second example? What series would we get if we let

$x=1$ ? or

$x=2$ ? Notice that the

$x$ is the same in every term, while the coefficient may vary depending on

$n$ .

DO: Write, in summation notation as well as the first few terms, the power series with $c_n=1$ for all $n$ . Does this series converge at $x=\frac{3}{4}$ ? How about at $x={1.2}$ ?

DO: Write, in summation notation as well as the first few terms, the power series beginning at $n=1$ , with $c_n=\frac{1}{n^2}$ for each $n$ . Does this series converge at $x={1}$ ? How about at $x=\frac{5}{4}$ ? Hint: remember your convergence tests!

We are interested in which values of $x$ will give us a convergent series. Such $x$ are like the domain of the power series, since the series makes sense at values of $x$ for which it converges, and makes no sense at other values of $x$ for which it is divergent. It turns out that the $x$ -values for which a power series converge consist of an interval, with the center of the interval being the center of the power series. (Here, as mentioned above but not explained, we are looking at series centered at the origin.) We will discuss this in more detail as we proceed.

Definitions (for a power series, centered at the origin)
The set of

$x$ where the series converges is called the interval of convergence, and is an interval from

$-R$ to

$R$ , where

$R\ge0$ is a called the radius of convergence.

The interval of convergence may include one, both, or no endpoints. Special cases will be when $R=0$ , where the series converges at the single point $x=0$ , and $R=\infty$ , where the series converges on the entire real line. In notation, the possible intervals of convergence for a series centered at the origin are

$[-R,R],\quad [-R,R), \quad(-R,R], \quad(-R,R),\quad\{0\}=[0,0],\quad\text{ and }\quad(-\infty,\infty).$

The following video will discuss these concepts and consider some examples.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Power Series: Radius and Interval of Convergence