Power Series

Q: What is the difference between a power series and a Taylor series?

A power series is any series of the form $\sum a_n(x-c)^n$ with arbitrary coefficients. A Taylor series is a specific power series where the coefficients are determined by the derivatives of a known function: $a_n = \frac{f^{(n)}(c)}{n!}$. Every Taylor series is a power series, but not every power series arises as the Taylor series of a function.

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

The most common method is the ratio test. Compute $L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$. Then the radius of convergence is $R = \frac{1}{L}$. If $L = 0$, the series converges for all $x$ ($R = \infty$). If $L = \infty$, the series converges only at $x = c$ ($R = 0$). You must check the endpoints separately.

Power Series

A series which represents a function as a polynomial that goes on forever and has no highest power of x.

Power series formulas: sum c_k·x^k and sum c_k·(x-a)^k, with example e^x = sum x^k/k! = 1+x+x²/2!+x³/3!+x⁴/4!+…

Key Formula

\sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + \cdots

Where:

$a_n$ = The coefficient of the nth term, which may follow a pattern or formula
$x$ = The variable; the series defines a function of x
$c$ = The center of the power series (when c = 0, the series is centered at the origin)
$n$ = The index of summation, running from 0 to infinity

Worked Example

Problem: Write the power series representation of

f(x) = \frac{1}{1-x}

centered at

c = 0

, and determine for which values of

x

it converges.

Step 1: Recognize the geometric series formula. A geometric series with first term 1 and common ratio

x

sums to

\frac{1}{1-x}

when

|x| < 1

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n

Step 2: Expand the first several terms to see the pattern. Here every coefficient

a_n = 1

and the center

c = 0

\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots

Step 3: Apply the ratio test to find the radius of convergence. Compute the limit of the ratio of consecutive terms.

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

Step 4: The series converges when the ratio is less than 1, so

|x| < 1

. The radius of convergence is

R = 1

, and the interval of convergence is

(-1, 1)

|x| < 1 \implies R = 1

Answer: The power series representation is

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots

, valid for

|x| < 1

Another Example

This example uses factorial coefficients (unlike the constant coefficients in Example 1) and shows a power series with an infinite radius of convergence, meaning it converges everywhere.

Problem: Find the power series representation of

e^x

centered at

c = 0

and verify the first four terms by evaluating at

x = 1

Step 1: The Maclaurin series (power series centered at 0) for

e^x

uses the fact that every derivative of

e^x

evaluated at 0 equals 1. So

a_n = \frac{1}{n!}

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

Step 2: Write out the first four terms explicitly.

e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots

Step 3: Substitute

x = 1

into the first four terms to approximate

e^1 = e

1 + 1 + \frac{1}{2} + \frac{1}{6} = \frac{8}{3} \approx 2.6667

Step 4: Compare with the actual value

e \approx 2.7183

. The four-term approximation is already close. Adding more terms brings the sum closer to

e

\text{Error} \approx 2.7183 - 2.6667 = 0.0516

Step 5: Determine convergence. The ratio test gives

\lim_{n\to\infty} \frac{|x|}{n+1} = 0

for all

x

, so this power series converges for every real number. The radius of convergence is

R = \infty

R = \infty

Answer: The power series is

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

, converging for all real

x

. At

x=1

, four terms give approximately 2.667, close to

e \approx 2.718

Frequently Asked Questions

What is the difference between a power series and a Taylor series?

A power series is any series of the form

\sum a_n(x-c)^n

with arbitrary coefficients. A Taylor series is a specific power series where the coefficients are determined by the derivatives of a known function:

a_n = \frac{f^{(n)}(c)}{n!}

. Every Taylor series is a power series, but not every power series arises as the Taylor series of a function.

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

The most common method is the ratio test. Compute

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

. Then the radius of convergence is

R = \frac{1}{L}

. If

L = 0

, the series converges for all

x

(

R = \infty

). If

L = \infty

, the series converges only at

x = c

(

R = 0

). You must check the endpoints separately.

When do you use a power series?

Power series are used to approximate functions that are difficult to compute exactly, to solve differential equations, and to evaluate integrals that have no closed-form antiderivative. For instance, the series for

e^x

\sin x

, and

\ln(1+x)

are standard tools in calculus, physics, and engineering.

Power Series vs. Taylor Series

	Power Series	Taylor Series
Definition	Any infinite series $\sum a_n(x-c)^n$ with given coefficients	A power series whose coefficients come from derivatives of a specific function
Formula	$\sum_{n=0}^{\infty} a_n(x-c)^n$	$\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
Coefficients	Can be any sequence of numbers	Determined by $a_n = f^{(n)}(c)/n!$
Center at 0	Power series centered at 0	Called a Maclaurin series
When to use	General framework for representing functions as infinite polynomials	When you know a function and want to expand it around a specific point

Why It Matters

Power series appear throughout calculus, physics, and engineering whenever you need to express a complicated function as an infinite polynomial. In AP Calculus BC and university courses, you will use them to approximate functions like

\sin x

and

e^x

, solve differential equations that resist other methods, and compute definite integrals that have no elementary antiderivative. Understanding power series also lays the foundation for Fourier series and other advanced representations of functions.

Common Mistakes

Mistake: Forgetting to check convergence at the endpoints of the interval.

Correction: The ratio test determines the open interval

|x - c| < R

, but the series may or may not converge at

x = c - R

and

x = c + R

. You must test each endpoint individually using another convergence test (e.g., alternating series test, p-series test).

Mistake: Assuming a power series converges everywhere just because it represents a well-known function.

Correction: Many functions have power series that converge only within a finite radius. For example,

\frac{1}{1-x} = \sum x^n

diverges for

|x| \geq 1

even though

\frac{1}{1-x}

is defined for all

x \neq 1

. Always compute the radius of convergence.

Related Terms

Taylor Series — Power series with coefficients from derivatives of a function
Maclaurin Series — Taylor series centered at zero
Radius of Convergence — Determines where a power series converges
Power Series Convergence — Tests and criteria for when the series sums to a finite value
Convergence Tests — Methods like ratio test used to find convergence
Derivative of a Power Series — Differentiate term by term within the radius
Integral of a Power Series — Integrate term by term within the radius
Series — General concept of summing infinite sequences