4.1: Power Series and Functions

Form of a Power Series

Definition: Power Series

A series of the form\[\sum_{n=0}^\infty c_n x^n = c_0+c_1x+c_2x^2+\cdots \nonumber \]is called a power series centered at \(x=0\). A series of the form\[\sum_{n=0}^\infty c_n(x−a)^n = c_0+c_1(x−a)+c_2(x−a)^2+\cdots \nonumber \]is a power series centered at \(x=a\). \( c_n \) are called the coefficients of the power series. The value(s) of \( x \) for which the series converges is called the interval of convergence. The largest value of \( R \) for which the series converges on \( \left( a - R, a + R \right) \) is called the radius of convergence.

For a visual of the interval of convergence, please see this link.

Note #1

A power series is a polynomial function of \( x \) with (quite possibly) an infinite amount of terms.

Note #2

This is the first example of a series that contains variables.

Convergence of a Power Series

Theorem: Convergence of a Power Series

The power series \(\displaystyle \sum_{n=0}^ \infty c_n(x−a)^n\) satisfies exactly one of the following properties:

1. The series converges at \(x=a\) and diverges for all \(x \neq a\).
2. The series converges for all real numbers \(x\).
3. There exists a real number \(R \gt 0\) such that the series converges if \(|x−a| \lt R\) and diverges if \(|x−a| \gt R\). At the values \(x\) where \(|x−a|=R\), the series could converge or diverge.

Proof (in the text)

Suppose that the power series is centered at \(a=0\). We must first prove the following fact:

If there exists a real number \(d \neq 0\) such that \(\displaystyle \sum_{n=0}^\infty c_n d^n\) converges, then the series \(\displaystyle \sum_{n=0}^ \infty c_nx^n\) converges absolutely for all \(x\) such that \(|x| \lt |d|\).

Suppose, then, that \(\displaystyle \sum_{n=0}^ \infty c_nd^n\) converges. Then the \(n^{\text{th}}\) term \(c_nd^n \to 0\) as \(n \to \infty \). Therefore, there exists an integer \(N\) such that \(|c_nd^n| \leq 1\) for all \(n \geq N\). Writing\[|c_nx^n|=|c_nd^n| \left|\dfrac{x}{d}\right|^n, \nonumber \]we conclude that, for all \(n \geq N\),\[|c_nx^n| \leq \left|\dfrac{x}{d}\right|^n. \nonumber \]

The series\[\sum_{n=N}^ \infty \left|\dfrac{x}{d}\right|^n \nonumber \]is a geometric series that converges if \(\left|\frac{x}{d}\right| \lt 1\). Therefore, by the Series Comparison Test, we conclude that \(\displaystyle \sum_{n=N}^ \infty c_n x^n\) also converges for \(|x| \lt |d|\). Since we can add a finite number of terms to a convergent series, we conclude that \(\displaystyle \sum_{n=0}^ \infty c_n x^n\) converges for \(|x| \lt |d|\).

With this result, we can now prove the theorem. Consider the series\[\sum_{n=0}^ \infty a_nx^n \nonumber \]and let \(S\) be the set of real numbers for which the series converges. Suppose that the set contains only the center. That is, \(S=\{0\}\). Then the series falls under case i.

Suppose that the set \(S\) is the set of all real numbers. Then the series falls under case ii.

Suppose that \(S \neq {0}\) and \(S\) is not the set of real numbers. Then there exists a real number \(x^* \neq 0\) such that the series does not converge. Thus, the series cannot converge for any \(x\) such that \(|x| \gt |x^*|\). Therefore, the set \(S\) must be a bounded set, meaning it must have a smallest upper bound. Call that smallest upper bound \(R\). Since \(S \neq \{0\}\), the number \(R \gt 0\). Therefore, the series converges for all \(x\) such that \(|x| \lt R\), and the series falls into case iii.

Note #3

All power series converge at their centers.

Note #4

We use our previous tests to determine when our power series converge (top choice is almost always the Ratio Test).

Note #5

Power series that are geometric-in-form never converge at the ends of their intervals of convergence. (Why?) For all other power series, you must test for the convergence at the endpoints separately.

Lecture Example \(\PageIndex{1}\)

For each of the following series, find the interval and radius of convergence.

1.    \[ \sum_{n=0}^ \infty \frac{x^n}{n!} \nonumber \](let's use Desmos to visualize this!)
2.    \[ \sum_{n=1}^ \infty \dfrac{(x - 2)^n}{n^n} \nonumber \]
3.    \[ \sum_{n=0}^ \infty \dfrac{(x−2)^n}{n^2 + 1} \nonumber \]
4.    \[ \sum_{n = 1}^{\infty} \dfrac{2^n}{n} (4x - 8)^n \nonumber \]
5.    \[ \sum_{n = 0}^{\infty} n! (2x + 1)^n \nonumber \]

One Power Series to Rule Them All

Theorem

\[ \sum_{n = 0}^{\infty} x^n = \dfrac{1}{1 - x}, \quad |x| < 1 \nonumber \]

Lecture Example \(\PageIndex{2}\)

Find a power series representation for the function and determine the interval of convergence.

1.    \[ f(x) = \dfrac{2}{9 + x} \nonumber \]
2.    \[ g(x) = \dfrac{2x^2}{1 + x^3} \nonumber \]