Representing a function as a power series means rewriting it as an infinite polynomial, usually by starting from a series you already know and transforming it. The fastest moves are multiplying or substituting into known series like $e^x$ , $\sin x$ , and $\cos x$ , plus differentiating or integrating a known series term by term. This is the final topic of AP Calculus BC and pulls together the series skills from the whole unit.

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How Do You Represent a Function as a Power Series?

To represent a function as a power series, start with a known series and transform it. Common moves include substituting a new expression for $x$ , multiplying by a power of $x$ , using the geometric series, or differentiating or integrating term by term inside the interval of convergence.

For AP Calculus BC, show both the first few nonzero terms and a correct general term. The first terms help reveal the pattern, and the general term proves the full power series representation.

Why This Matters for the AP Calculus Exam

Power series questions show up on both multiple-choice and free-response parts of the AP Calculus BC exam. The main skill is recognizing a function as a transformed version of a known series and then building the new series from it instead of computing every derivative from scratch.

This topic rewards procedure choice and clean notation. You often need to give the first few nonzero terms and the general term, so showing how you transformed a known series makes your work clear and easy to follow. Differentiating or integrating series term by term also connects directly to earlier Unit 10 ideas like interval of convergence.

Key Takeaways

A power series has the form $\displaystyle\sum_{n=0}^{\infty} a_n(x-r)^n$ , where $r$ is the center and $a_n$ is the sequence of coefficients.
Know these three by heart: $e^x$ , $\sin x$ , and $\cos x$ . Most problems start from one of them.
To build a new series, transform a known one: multiply by a power of $x$ , substitute something for $x$ , or use the geometric series $\frac{1}{1-x}=\sum x^n$ .
You can differentiate or integrate a power series term by term, which is often faster than finding each derivative directly.
When asked, give the first few nonzero terms and the general term, since the general term shows the full pattern.
Watch the interval of convergence when you substitute or shift, because it can change.

The Three Series to Memorize

Each of these is centered at $x = 0$ (a Maclaurin series). Knowing them lets you generate many other series by transforming.

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots+\frac{x^n}{n!}

\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots+\frac{(-1)^nx^{2n}}{(2n)!}

\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}

The geometric series is another go-to:

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \ldots \quad \text{for } |x| < 1

From here you can substitute or integrate to get series for things like $\frac{1}{1+x}$ , $\ln(1+x)$ , and $\arctan x$ .

How to Use This on the AP Calculus Exam

Multiplying a Known Series

Find the power series for $x^2 e^x$ . Include the first 4 nonzero terms and the general term.

Start from $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...$ Since the function is $e^x$ multiplied by $x^2$ , multiply the whole series by $x^2$ :

x^2e^x=x^2[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...]

=x^2+x^3+\frac{x^4}{2!}+\frac{x^5}{3!}+...+\frac{x^{n+2}}{n!}+...

Differentiating a Known Series

Let $h(x)$ be the power series centered at $x=0$ of $\cos(x)$ . Find $h'(x)$ , including the first 3 nonzero terms and the general term.

Start from $h(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...+\frac{(-1)^nx^{2n}}{(2n)!}+...$ Differentiate term by term:

h'(x)=-x+\frac{x^3}{3!}-\frac{x^5}{5!}+...+\frac{(-1)^nx^{2n-1}}{(2n-1)!}+...\quad (n\geq1)

Notice this is the negative of the $\sin(x)$ series, which matches the fact that the derivative of $\cos(x)$ is $-\sin(x)$ .

Free Response

The following is Question 6 from the 2022 AP Calculus BC examination administered by College Board. All credit to College Board.

The function $f$ is defined by the power series $f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+...+\frac{(-1)^nx^{2n+1}}{2n+1}+...$ for all real numbers $x$ for which the series converges.

Part (c): Write the first four nonzero terms and the general term for an infinite series that represents $f'(x)$ .

Differentiate the given series term by term, like in the previous example:

f'(x)=1-x^2+x^4-x^6+...+(-1)^nx^{2n}+...

Common Trap

When a problem asks for the general term, do not stop at the first few terms. The general term proves you found the pattern, and it is usually required for full credit on free-response work.

Common Misconceptions

A power series is not just a Taylor polynomial. The polynomial is a finite approximation; the power series is the full infinite sum.
Transforming a known series does not always keep the same interval of convergence. Substituting $x \to ax$ or shifting the center can change where the series converges, so check the new interval when it matters.
Differentiating or integrating term by term is valid inside the interval of convergence, but the behavior at the endpoints can still differ, so do not assume the endpoints carry over.
Multiplying a series by $x^2$ shifts every exponent up by 2; remember to adjust the general term's exponent (for example $x^n$ becomes $x^{n+2}$ ), not just the visible terms.
Memorizing $e^x$ , $\sin x$ , and $\cos x$ is worth it, but you still need to track signs and factorials carefully, since a single dropped sign changes the whole pattern.

As you review, check out the Study Tools unit for more resources and exam information.

Frequently Asked Questions

How do you represent a function as a power series?

Start with a known series and transform it by substitution, multiplication, the geometric series, term-by-term differentiation, or term-by-term integration. Then write the first few nonzero terms and a general term.

Which known series are most useful for power series representations?

The most useful starting points are e to the x, sine x, cosine x, and the geometric series one over one minus x. Many other power series come from substituting, differentiating, integrating, or multiplying these series.

When can you differentiate or integrate a power series term by term?

You can differentiate or integrate term by term inside the interval of convergence. The radius of convergence stays the same, but endpoint behavior may change and should be checked when needed.

Why do AP problems ask for first terms and a general term?

The first few terms show the pattern, while the general term states the full infinite series. On free-response questions, stopping at visible terms can lose credit if the problem asks for the full representation.

How does the geometric series help represent functions?

The geometric series one over one minus x equals the sum of x to the n for absolute value of x less than 1. By rewriting a function into that form, you can create power series for rational functions and then transform them further.

How is representing functions as power series tested on AP Calculus BC?

AP Calculus BC may ask you to transform a known series, differentiate or integrate a series, write a general term, or state an interval of convergence. Track signs, powers, factorials, and endpoints carefully.