Power Series

Main idea

A power series is a polynomial that never ends. Pick a value of $x$ and it turns into an ordinary series of numbers, so everything from the last unit applies, and it always works on an interval centered at $a$ and fails outside it.

\sum_{n=0}^{\infty} {\htmlData{sym=0}{c_n}} ({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}})^n = c_0 + c_1({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}}) + c_2({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}})^2 + \cdots

What the symbols mean

$c_n$
Coefficients
Fixed numbers, one for each power.
$a$
Center
The series is built around x = a, where it always converges.
$x$
Variable
Plugging in a value of x gives an ordinary series of numbers.

Radius of convergence

|x - a| < {\htmlData{sym=0}{R}} \implies \text{converges}, \qquad |x - a| > {\htmlData{sym=0}{R}} \implies \text{diverges}

What the symbols mean

$R$
Radius of convergence
Half the width of the interval where the series converges. It can be 0 or ∞.

Explanation

A power series is simply a polynomial that keeps going forever, adding higher and higher powers of $x$ . (The "power" is those powers of $x$ ; the "series" is the endless sum.) They're useful because polynomials are easy to compute, and it turns out functions like $e^x$ , $\sin x$ and $\ln x$ can be written this way. $\sum c_n (x-a)^n$ looks abstract, but it's just $c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$ : $c_n$ (an ordinary number in front of each power), $a$ (the center, the spot the series is built around), and $(x - a)^n$ (how far $x$ is from that center, raised to higher and higher powers). The only new question is which values of $x$ make the endless sum add up to a finite number. That's it; bear with the notation.

You've actually met one already. The geometric series $1 + x + x^2 + \cdots$ is a power series with every $c_n = 1$ , and you know it adds up to $\frac{1}{1-x}$ when $|x| < 1$ and blows up otherwise. Every power series behaves a lot like that.

Plug in a number for $x$ and a power series becomes a series of numbers, which either converges or doesn't. The set of $x$ where it converges is the interval of convergence. There are only three possibilities:

It converges only at $x = a$ ( $R = 0$ ).
It converges for every $x$ ( $R = \infty$ ).
It converges when $|x - a| < R$ and diverges when $|x - a| > R$ , for some number $R > 0$ .

Think about why it has to be symmetric around $a$ . The terms are powers of $(x - a)$ , and for big $n$ what decides whether they shrink is the size $|x - a|$ , not which side of $a$ you're on. So the series can't tell $a + 2$ from $a - 2$ , at least away from the edges.

Finding $R$ . Apply the ratio test (or the root test) with $x$ left as a letter. You'll get a limit like $L = |x - a|\cdot k$ . The series converges when $L < 1$ , which means $|x - a| < 1/k$ , so $R = 1/k$ .

Endpoints. At $x = a \pm R$ the ratio test gives $L = 1$ and says nothing. So plug each endpoint in and test the resulting series by itself, usually with the p-series or alternating series test. This is the one place the two sides can behave differently, so the interval can be open, closed, or half-open.

See it

These are partial sums of 1 + x + x² + ⋯ laid over the function they're trying to become, 1/(1 − x). As you add terms, they hug the curve more and more closely between −1 and 1. Past those lines they swing wildly and never settle, no matter how many terms you add. That boundary is the radius of convergence, R = 1.

Worked examples

Example 1

Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ .

1
Ratio test: $\left|\dfrac{x^{n+1}}{n+1}\cdot\dfrac{n}{x^n}\right| = |x|\,\dfrac{n}{n+1} \to |x|$ . So $R = 1$ .

Example 2

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} n!\,(x - 2)^n$ .

1
$\left|\dfrac{(n+1)!\,(x-2)^{n+1}}{n!\,(x-2)^n}\right| = (n+1)|x - 2| \to \infty$ unless $x = 2$ .

$R = 0$ : it converges only at $x = 2$ .

Example 3

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$ .

1
$\left|\dfrac{x^{n+1}}{(n+1)!}\cdot\dfrac{n!}{x^n}\right| = \dfrac{|x|}{n+1} \to 0 < 1$ for every $x$ .

$R = \infty$ . This one is the series for $e^x$ , as we'll see in the Taylor series lesson.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the radius of convergence of $\displaystyle\sum \frac{2^n x^n}{n}$ .

Problem 2+5 XP

Find the radius of convergence of $\displaystyle\sum \frac{(x - 2)^n}{5^n\sqrt n}$ .

Problem 3+5 XP

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n^2\, 7^n}$ ?

Problem 4+5 XP

Find the radius of convergence of $\displaystyle\sum n!\,x^n$ .

Problem 5+5 XP

The series $\displaystyle\sum_{n=0}^{\infty} \frac{(x+1)^n}{3^n}$ converges on an open interval. Give both endpoints.

Problem 6+5 XP

An echo comes back with its loudness multiplied by $\frac x3$ each time it bounces, so the total sound is $\sum_{n=0}^\infty\left(\frac x3\right)^n$ times the original. For which values of $x$ is the total finite? Give the radius of convergence.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the interval of convergence of $\displaystyle\sum_{n=1}^\infty\frac{(x - 1)^n}{n\,3^n}$ , including the endpoints.

Where it's used

Calculators, physics engines and graphics shaders compute functions with a few terms of a power series. The radius of convergence says how far from the center the approximation can be trusted.

Taylor SeriesCalc II · Computer Science

Estimating the Value of a Series Power Series and Functions

Power Series

Main idea

\sum_{n=0}^{\infty} {\htmlData{sym=0}{c_n}} ({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}})^n = c_0 + c_1({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}}) + c_2({\htmlData{sym=2}{x}}-{\htmlData{sym=1}{a}})^2 + \cdots

What the symbols mean

$c_n$
Coefficients
Fixed numbers, one for each power.
$a$
Center
The series is built around x = a, where it always converges.
$x$
Variable
Plugging in a value of x gives an ordinary series of numbers.

Radius of convergence

|x - a| < {\htmlData{sym=0}{R}} \implies \text{converges}, \qquad |x - a| > {\htmlData{sym=0}{R}} \implies \text{diverges}

What the symbols mean

$R$
Radius of convergence
Half the width of the interval where the series converges. It can be 0 or ∞.

Explanation

It converges only at $x = a$ ( $R = 0$ ).
It converges for every $x$ ( $R = \infty$ ).
It converges when $|x - a| < R$ and diverges when $|x - a| > R$ , for some number $R > 0$ .

See it

Worked examples

Example 1

Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ .

1
Ratio test: $\left|\dfrac{x^{n+1}}{n+1}\cdot\dfrac{n}{x^n}\right| = |x|\,\dfrac{n}{n+1} \to |x|$ . So $R = 1$ .

Example 2

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} n!\,(x - 2)^n$ .

1
$\left|\dfrac{(n+1)!\,(x-2)^{n+1}}{n!\,(x-2)^n}\right| = (n+1)|x - 2| \to \infty$ unless $x = 2$ .

$R = 0$ : it converges only at $x = 2$ .

Example 3

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$ .

1
$\left|\dfrac{x^{n+1}}{(n+1)!}\cdot\dfrac{n!}{x^n}\right| = \dfrac{|x|}{n+1} \to 0 < 1$ for every $x$ .

$R = \infty$ . This one is the series for $e^x$ , as we'll see in the Taylor series lesson.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the radius of convergence of $\displaystyle\sum \frac{2^n x^n}{n}$ .

Problem 2+5 XP

Find the radius of convergence of $\displaystyle\sum \frac{(x - 2)^n}{5^n\sqrt n}$ .

Problem 3+5 XP

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n^2\, 7^n}$ ?

Problem 4+5 XP

Find the radius of convergence of $\displaystyle\sum n!\,x^n$ .

Problem 5+5 XP

The series $\displaystyle\sum_{n=0}^{\infty} \frac{(x+1)^n}{3^n}$ converges on an open interval. Give both endpoints.

Problem 6+5 XP

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the interval of convergence of $\displaystyle\sum_{n=1}^\infty\frac{(x - 1)^n}{n\,3^n}$ , including the endpoints.

Where it's used

Calculators, physics engines and graphics shaders compute functions with a few terms of a power series. The radius of convergence says how far from the center the approximation can be trusted.

Taylor SeriesCalc II · Computer Science