A Taylor series writes a function as an infinite power series built from its derivatives at a center point: $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ . A Maclaurin series is just a Taylor series centered at $x=0$ . The fastest way to find these on the AP Calculus BC exam is to memorize a few common Maclaurin series and then substitute, multiply, differentiate, or integrate to match the function you are given.

more resources to help you study

practice multiple choice FRQ practice & scoring cheatsheets score calculator key terms

How Do You Find a Taylor or Maclaurin Series?

To find a Taylor or Maclaurin series, start from the derivative formula $\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$ or transform a known series. A Maclaurin series is the special case where the center is $a=0$ .

For AP Calculus BC, known series are usually faster than repeated differentiation. Use the geometric series for $\frac{1}{1-x}$ and memorize the Maclaurin series for $e^x$ , $\sin x$ , and $\cos x$ , then substitute, multiply, differentiate, or integrate to match the function.

Why This Matters for the AP Calculus Exam

This is a BC-only topic, so you will only see it if you take AP Calculus BC. It pulls together two earlier ideas: approximating functions with Taylor polynomials and representing functions as power series. On the exam, series questions show up in both multiple-choice and free-response settings, and being able to recognize a function's series form quickly saves time. You will be expected to build a Taylor or Maclaurin series, list out terms, and reason about how known series connect to new ones.

Key Takeaways

The Taylor series of $f$ centered at $x=a$ is $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ , where $f^{(0)}(a)=f(a)$ .
A Maclaurin series is a Taylor series with center $a=0$ .
A Taylor polynomial is a partial sum (finite piece) of the full Taylor series.
Memorize the Maclaurin series for $\frac{1}{1-x}$ , $e^x$ , $\sin x$ , and $\cos x$ ; most other series come from these by substitution or algebra.
The series for $\frac{1}{1-x}$ is a geometric series, and the series for $\sin x$ , $\cos x$ , and $e^x$ are the building blocks for many other functions.
Pattern recognition in derivatives is the key skill when you have to build a series from scratch.

What's a Taylor Series?

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It lets you approximate functions and estimate their values near that point.

Taylor Series: For a function $f(x)$ , its Taylor series centered at $x = a$ is:

\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}\cdot(x-a)^n=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n

Here $f^{(n)}(a)$ is the $n^{\text{th}}$ derivative of the function evaluated at $a$ , and $f^{(0)}(a)=f(a)$ .

When the center is $x = 0$ , this special case is called a Maclaurin series. Taylor series centered at $0$ come up so often that they get their own name.

A Taylor polynomial for $f(x)$ is a partial sum of the Taylor series for $f(x)$ . In other words, a Taylor polynomial is a finite polynomial with a limited number of terms, while a Taylor series is an infinite summation of terms. To build a Taylor polynomial of degree $n$ for $f(x)$ at $x = c$ , evaluate $f$ and its first $n$ derivatives at $x = c$ .

To form the full Taylor series, it helps to find a pattern that describes the $n$ th derivative of $f$ at $x = c$ . Pattern recognition is the most useful skill here.

Important Maclaurin Series to Remember

Focus on Maclaurin series (center $x = 0$ ). If a problem gives you a different center and you already know the Maclaurin form, you can replace $x$ with $(x - c)$ in each term and simplify.

\begin{array}{ |c|c|c| } \hline f(x)& Series \enspace Representation & Expanded \enspace Form \\\\ \frac{1}{1-x} & \sum_{n=0}^\infty{x^n} & 1+x+x^2+x^3+... \\\\ \frac{1}{1+x} & \sum_{n=0}^\infty{(-x)^n} & 1-x+x^2-x^3+... \\\\ \frac{1}{1+{x}^2} & \sum_{n=0}^\infty{(-x)^{2n}} & 1-x^2+x^4-x^6+... \\\\ \frac{1}{1-{x}^2} & \sum_{n=0}^\infty{x^{2n}} & 1+x^2+x^4+x^6+... \\\\ e^x & \sum_{n=0}^\infty\frac{x^n}{n!} & 1+x+\frac{x^2}{2}+\frac{x^3}{3!}+... \\\\ sin(x) & \sum_{n=0}^\infty{(-1)^n\frac{x^{2n+1}}{(2n+1)!}} & x-\frac{x^3}{3!}+\frac{x^5}{5!}+... \\\\ cos(x) & \sum_{n=0}^\infty{(-1)^n\frac{x^{2n}}{(2n)!}} & 1-\frac{x^2}{2}+\frac{x^4}{4!}+... \\\\ ln(1+x) & \sum_{n=1}^\infty{(-1)^{n+1}\frac{x^{n}}{n}} & x-\frac{x^2}{2}+\frac{x^3}{3}+... \\\\ (1+x)^a & \sum_{n=0}^\infty{a \choose n}x^n & 1+ax+\frac{a(a-1)}{2}x^2+\frac{a(a-1)(a-2)}{3!}x^3+... \\\\ \hline \end{array}

Two things to notice:

The first four are variations of the geometric series, with the multiplying factor being various powers of $x$ . The series $(1+x)^a$ is the binomial series for any value of $a$ .
The Maclaurin series for $\sin(x)$ , $\cos(x)$ , and $e^x$ are the foundation for building the Maclaurin series of many other functions.

Looking at point (2), the patterns connect: $\sin(x)$ is an odd function, so its series has only odd powers, while $\cos(x)$ is even, so its series has only even powers. The series for $e^x$ uses all powers. If you are short on time, $\sin(x)$ , $\cos(x)$ , and $e^x$ are the must-know series.

How to Use This on the AP Calculus Exam

Problem Solving

Start by checking whether the function matches a known Maclaurin series. If it does, substitution is usually faster than computing derivatives one by one.

Question 1. Find the Taylor series for $f(x) = \cos(3x)$ centered at $x = 0$ .

Centered at $x = 0$ means this is a Maclaurin series. Pull the series for $\cos(x)$ :

\sum_{n=0}^\infty{(-1)^n\frac{x^{2n}}{(2n)!}}

Replace $x$ with $3x$ and simplify:

\sum_{n=0}^\infty{(-1)^n\frac{(3x)^{2n}}{(2n)!}}=\sum_{n=0}^\infty{(-1)^n\frac{3^{2n}x^{2n}}{(2n)!}}=\sum_{n=0}^\infty{(-1)^n\frac{9^{n}x^{2n}}{(2n)!}}

That is the full answer. The more comfortable you are recalling common Maclaurin series, the faster these go.

Question 2a. Find the Taylor series centered at $x = 5$ for $f(x)=e^{2x}$ .

Compute a few derivatives and look for a pattern:

f(x) = e^{2x}

f'(x) = 2e^{2x}

f''(x) = 4e^{2x} = (2)^2e^{2x}

f^{(3)}(x) = 8e^{2x} = (2)^3e^{2x}

f^{(4)}(x) = 16e^{2x} = (2)^4e^{2x}

The $n$ th derivative brings out a factor of $2^n$ . Using the structure of the $e^x$ series:

\sum_{n=0}^\infty\frac{x^n}{n!} \Rightarrow \sum_{n=0}^\infty\frac{f^{(n)}(5)}{n!}(x-5)^n

The center is $x = 5$ , so write $(x-5)^n$ instead of $x^n$ and evaluate the $e^{2x}$ factor at $x = 5$ . Leave the $n$ exponents alone:

\sum_{n=0}^\infty\frac{f^{(n)}(5)}{n!}(x-5)^n=\sum_{n=0}^\infty\frac{2^ne^{10}(x-5)^n}{n!}

Question 2b. List the first four terms of that series.

Plug in $n = 0, 1, 2, 3$ :

\begin{array}{ |c|c| } \hline n & Term \\\\ 0 & \frac{(2)^0e^{10}(x-5)^0}{0!}=e^{10} \\\\ 1 & \frac{(2)^1e^{10}(x-5)^1}{1!} =2e^{10}(x-5)\\\\ 2 & \frac{(2)^2e^{10}(x-5)^2}{2!} =\frac{4e^{10}(x-5)^2}{2}=2e^{10}(x-5)^2\\\\ 3 & \frac{(2)^3e^{10}(x-5)^3}{3!}=\frac{8e^{10}(x-5)^3}{3*2*1}=\frac{4e^{10}(x-5)^3}{3} \\\\ \hline \end{array}

Common Trap

When you shift a known Maclaurin series to a new center, change only the $(x-c)$ base and any constant factors that depend on $c$ . Do not change the $n$ in the exponents or factorials.

Common Misconceptions

A Taylor polynomial and a Taylor series are not the same. The polynomial is a finite partial sum; the series is the full infinite sum.
A Maclaurin series is not a separate type of series. It is just a Taylor series with center $a = 0$ .
When you substitute something like $3x$ into a known series, replace $x$ everywhere it appears, including inside exponents, then simplify (for example $(3x)^{2n} = 9^n x^{2n}$ ).
Changing the center to $x = c$ means replacing $x$ with $(x-c)$ in the power terms, not adding $c$ to the index $n$ .
Knowing the series for $\sin x$ , $\cos x$ , and $e^x$ does not replace pattern recognition. When a function does not match a known form, you still need to differentiate and find the pattern in $\frac{f^{(n)}(a)}{n!}$ .

Vocabulary

The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.

Term	Definition
geometric series	A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}.
Maclaurin series	A special case of a Taylor series where the function is expanded around the point x = 0.
Taylor polynomial	A finite polynomial that approximates a function, formed by taking a partial sum of the Taylor series for that function.
Taylor series	A power series representation of a function that converges to that function over an open interval with positive radius of convergence.

Frequently Asked Questions

How do you find a Taylor or Maclaurin series?

Use the Taylor series derivative formula or transform a known series. A Maclaurin series is centered at 0, so the known series for e to the x, sine, cosine, and one over one minus x are often the fastest starting points.

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a Taylor series centered at 0. A Taylor series can be centered at any value a and uses powers of x minus a.

Which Maclaurin series should AP Calculus BC students memorize?

Know the series for one over one minus x, e to the x, sine x, and cosine x. The AP CED specifically names the geometric series and the Maclaurin series for sine, cosine, and e to the x as foundations for other functions.

How do substitutions work in Taylor and Maclaurin series?

Replace x in the known series with the new expression, then simplify powers, signs, and coefficients. For example, substituting 3x into cosine changes x to 3x everywhere in the cosine series.

What is the relationship between Taylor polynomials and Taylor series?

A Taylor polynomial is a partial sum of the full Taylor series. Topic 10.14 focuses on building the infinite series, while earlier work with Taylor polynomials uses finite approximations.

How is finding Taylor or Maclaurin series tested on AP Calculus BC?

AP Calculus BC may ask for the first few terms, a general term, or a transformed known series. Show the known series or derivative pattern and keep signs, factorials, and powers organized.