Radius and interval of convergence of a power series

Warm-up

Three quick questions from earlier lessons.

Discover

$\displaystyle\sum_{n=0}^{\infty} x^n$ is geometric. For which $x$ does it converge?

$-1 < x < 1$
$-1 \le x \le 1$

Key idea

A power series converges in an interval around its center.

\displaystyle\sum c_n (x - a)^n

converges for

|x - a| < R

and diverges for

|x - a| > R

(the radius). Find

R

with the ratio test. At the endpoints

x = a \pm R

anything can happen: test each one separately.

Worked example

$\displaystyle\sum_{n=1}^{\infty} \dfrac{x^n}{n\,3^n}$ .

R = 3

Ratio test: |x|/3 · n/(n + 1) → |x|/3.

Right endpoint excluded

x = 3: ∑1/n diverges.

Interval

[-3, 3)

x = −3: ∑(−1)ⁿ/n converges.

Your turn, guided

Your turn: the radius of $\displaystyle\sum \dfrac{x^n}{5^n}$ .

R = 5

|x|/5 < 1.

Practice ★

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 8^n}$ .
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 3^n}$ .

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} (2x)^n$ .
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} (5x)^n$ .

Watch out!

Practice ★★

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 7)^n}{n}$ ?
1. $[6, 8)$
2. $(6, 8)$
3. $[6, 8]$

Exit ticket

Three questions to check you have got it.

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 8^n}$ .
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 3^n}$ .
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 9^n}$ .

Summary card

Radius and interval

Ratio test → $|x - a| < R$ .
Test $x = a - R$ and $x = a + R$ separately.

Watch out: solve for $|x - a|$ carefully; never skip the endpoints.

End of the lesson

Next lesson: Maclaurin series of eˣ, sin x, cos x and 1/(1 − x) Back to the chapter

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