Mathwords: Power Series Convergence

Q: How do you find the radius of convergence of a power series?

The most common method is the Ratio Test: compute $R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$ where $a_n$ are the coefficients. Alternatively, use the Root Test: $1/R = \lim_{n \to \infty} |a_n|^{1/n}$. Both methods give the radius $R$, but neither tells you what happens at the endpoints, which you must check separately using other convergence tests.

Q: What is the difference between radius of convergence and interval of convergence?

The radius of convergence $R$ is a single non-negative number (or $\infty$) that measures how far from the center the series converges. The interval of convergence is the actual set of $x$-values where the series converges, including or excluding the endpoints as determined by separate tests. For example, a series with $R = 3$ centered at 0 could have an interval of $(-3, 3)$, $[-3, 3)$, $(-3, 3]$, or $[-3, 3]$.

Key Formula

\sum_{n=0}^{\infty} a_n (x - c)^n \quad \text{converges absolutely when } |x - c| < R \text{ and diverges when } |x - c| > R

Where:

$a_n$ = The coefficient of the nth term in the power series
$x$ = The variable
$c$ = The center of the power series
$R$ = The radius of convergence (0, a finite positive number, or ∞)
$n$ = The index of summation, starting from 0

Worked Example

Problem: Determine the radius and interval of convergence for the power series

\sum_{n=0}^{\infty} \frac{x^n}{3^n}

.

Step 1: Identify the general term of the series. Here

a_n = \frac{1}{3^n}

and the center is

c = 0

.

\sum_{n=0}^{\infty} \frac{x^n}{3^n} = \sum_{n=0}^{\infty} \left(\frac{x}{3}\right)^n

Step 2: Apply the Ratio Test. Compute the limit of the absolute ratio of consecutive terms.

L = \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{3} \right| = \frac{|x|}{3}

Step 3: The series converges absolutely when

L < 1

, so set up the inequality.

\frac{|x|}{3} < 1 \implies |x| < 3

Step 4: The radius of convergence is

R = 3

. Now check the endpoints

x = -3

and

x = 3

.

x = 3: \sum_{n=0}^{\infty} 1^n = \infty \quad (\text{diverges}), \qquad x = -3: \sum_{n=0}^{\infty} (-1)^n \quad (\text{diverges})

Step 5: Since the series diverges at both endpoints, the interval of convergence is the open interval.

(-3, 3)

Answer: The radius of convergence is

R = 3

and the interval of convergence is

(-3, 3)

.

Another Example

This example differs because the series is centered at $c = 2$ instead of 0, and the endpoints have different convergence behavior — one converges conditionally and the other diverges. This shows that you must always test endpoints individually.

Problem: Determine the radius and interval of convergence for

\sum_{n=1}^{\infty} \frac{(x - 2)^n}{n}

.

Step 1: Identify the coefficients and center. Here

a_n = \frac{1}{n}

and the series is centered at

c = 2

.

\sum_{n=1}^{\infty} \frac{(x-2)^n}{n}

Step 2: Apply the Ratio Test to find the radius of convergence.

L = \lim_{n \to \infty} \left| \frac{(x-2)^{n+1}}{n+1} \cdot \frac{n}{(x-2)^n} \right| = |x - 2| \cdot \lim_{n \to \infty} \frac{n}{n+1} = |x - 2|

Step 3: Set

L < 1

to find the radius.

|x - 2| < 1 \implies R = 1

Step 4: Check the left endpoint

x = 1

. Substituting gives an alternating harmonic series, which converges by the Alternating Series Test.

x = 1: \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \quad (\text{converges conditionally})

Step 5: Check the right endpoint

x = 3

. This gives the harmonic series, which diverges.

x = 3: \sum_{n=1}^{\infty} \frac{1}{n} \quad (\text{diverges})

Answer: The radius of convergence is

R = 1

and the interval of convergence is

[1, 3)

.

Frequently Asked Questions

What are the three cases of power series convergence?

Every power series satisfies exactly one of these: (1) it converges only at its center

x = c

, meaning

R = 0

; (2) it converges for all real numbers, meaning

R = \infty

; or (3) there exists a finite positive number

R

such that the series converges absolutely for

|x - c| < R

and diverges for

|x - c| > R

. In the third case, convergence at the endpoints

x = c \pm R

must be checked separately.

How do you find the radius of convergence of a power series?

The most common method is the Ratio Test: compute

R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|

where

a_n

are the coefficients. Alternatively, use the Root Test:

1/R = \lim_{n \to \infty} |a_n|^{1/n}

. Both methods give the radius

R

, but neither tells you what happens at the endpoints, which you must check separately using other convergence tests.

What is the difference between radius of convergence and interval of convergence?

The radius of convergence

R

is a single non-negative number (or

\infty

) that measures how far from the center the series converges. The interval of convergence is the actual set of

x

-values where the series converges, including or excluding the endpoints as determined by separate tests. For example, a series with

R = 3

centered at 0 could have an interval of

(-3, 3)

,

[-3, 3)

,

(-3, 3]

, or

[-3, 3]

.

Radius of Convergence vs. Interval of Convergence

	Radius of Convergence	Interval of Convergence
What it is	A single number $R \geq 0$ (or $\infty$ )	A set of all $x$ -values where the series converges
How to find it	Use the Ratio Test or Root Test on the coefficients	Find $R$ first, then test endpoints individually
Endpoint information	Does not specify endpoint behavior	Includes or excludes each endpoint based on testing
Example result	$R = 3$	$(-3, 3]$ or $[-3, 3)$ , etc.

Why It Matters

Power series convergence is central to calculus II and real analysis courses. You need it whenever you represent functions as Taylor or Maclaurin series — the radius of convergence tells you exactly where that representation is valid. It also appears in differential equations, physics, and engineering whenever solutions are expressed as infinite series expansions.

Common Mistakes

Mistake: Forgetting to check the endpoints separately after finding the radius of convergence.

Correction: The Ratio Test and Root Test are inconclusive when

|x - c| = R

. You must substitute each endpoint into the series and apply a different test (such as the Alternating Series Test, p-test, or comparison test) to determine whether it converges or diverges at that point.

Mistake: Confusing absolute convergence within the radius with convergence at the endpoints.

Correction: Inside the interval

|x - c| < R

, the series converges absolutely. At the endpoints, the series may converge absolutely, converge conditionally, or diverge. These are three distinct possibilities that require individual analysis.

Related Terms

Power Series — The type of series whose convergence is analyzed
Radius of Convergence — The value $R$ determining where the series converges
Interval of Convergence — The full set of $x$ -values including endpoint analysis
Convergence Tests — Methods used to determine convergence behavior
Convergent Series — A series whose partial sums approach a finite limit
Divergent Series — A series that does not converge
Theorem — Power series convergence is stated as a theorem