---
title: "Finding Taylor and Maclaurin Series | AP Calculus BC 10.14"
description: "Review how to find Taylor and Maclaurin series for AP Calculus BC, including known series, geometric series, substitution, derivative patterns, and general terms."
canonical: "https://fiveable.me/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/finding-taylor-or-maclaurin-series-for-function/study-guide/aKEvYorayYkUSTp1eCXv"
type: "study-guide"
subject: "AP Calculus AB/BC"
unit: "Unit 10 – Infinite Sequences and Series (BC Only)"
lastUpdated: "2026-06-08"
---

# Finding Taylor and Maclaurin Series | AP Calculus BC 10.14

## Summary

Review how to find Taylor and Maclaurin series for AP Calculus BC, including known series, geometric series, substitution, derivative patterns, and general terms.

## Guide

## TLDR
A [Taylor series](/ap-calc/key-terms/taylor-series "fv-autolink") writes a [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") 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.

## 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](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB "fv-autolink"). 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](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/working-with-geometric-series/study-guide/YvDdN4qbMyMaLgnFApU9 "fv-autolink") 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](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/representing-functions-as-power-series/study-guide/HsFR9pocBB6FdTMaZ8q7 "fv-autolink"). 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](/ap-calc/key-terms/partial-sum "fv-autolink") (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](/ap-calc/unit-6/integrating-functions-using-long-division-completing-square/study-guide/ju79RFY6f5aKWjFK "fv-autolink") 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](/ap-calc/unit-2/estimating-derivatives-function-at-point/study-guide/EzDxIj2eOHj3Ysb68Ew5 "fv-autolink") 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:

1. 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 $a$.
2. 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 $\frac{f^{(n)}(a)}{n!}$.

## Related AP Calculus Guides

- [Unit 10 Overview: Infinite Series and Sequences](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/review/study-guide/8ol6j4eNEB6GkkametRt)
- [10.1 Defining Convergent and Divergent Infinite Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB)
- [10.3 The nth Term Test for Divergence](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx)
- [10.5 Harmonic Series and p-Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/harmonic-series-p-series/study-guide/oaZ3mNFv3b8qBcsWmwIK)
- [10.2 Working with Geometric Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/working-with-geometric-series/study-guide/YvDdN4qbMyMaLgnFApU9)
- [10.4 Integral Test for Convergence](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/integral-test-for-convergence/study-guide/KrBj7QZJaHcPOKsThiS2)

## Vocabulary

- **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.
- **geometric series**: A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}.

## FAQs

### 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.

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