Finding Taylor or Maclaurin Series for a Function
Introduction: Why do we care about infinite polynomials? 🚀
students, imagine trying to work with a complicated function like $e^x$, $\sin x$, or $\ln(1+x)$ using only algebraic tools. One of the most powerful ideas in AP Calculus BC is that many functions can be written as infinite sums called Taylor series. These series let us approximate hard functions with polynomials, which are much easier to calculate, graph, and analyze.
The main goal of this lesson is to learn how to find a Taylor series, especially a Maclaurin series, for a function. A Maclaurin series is a Taylor series centered at $x=0$. This topic connects directly to infinite sequences and series because a Taylor series is itself an infinite series, and its usefulness depends on whether it converges to the original function.
Learning objectives
- Explain the meaning of Taylor and Maclaurin series.
- Find series for common functions using known basic series.
- Build new series from algebra, substitution, differentiation, and integration.
- Recognize when a series is centered at $x=0$ and when it is centered elsewhere.
- Connect series representations to approximation and error.
What is a Taylor series?
A Taylor series is a way to represent a function as an infinite polynomial centered at a value $a$. The general Taylor series for a function $f$ is
$$
$\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$
$$
Here, $f^{(n)}(a)$ means the $n$th derivative of $f$ evaluated at $x=a$. The number $a$ is the center of the series.
When the center is $a=0$, the series is called a Maclaurin series:
$$
$\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$
$$
This formula is the “official” construction, but on AP Calculus BC you often find series faster by using known parent series instead of computing every derivative from scratch. That saves time and reduces mistakes ✅
For example, the Maclaurin series for $e^x$ is
$$
$\displaystyle e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$
$$
This is one of the most important series in calculus because it is used as a starting point for many other expansions.
The basic Maclaurin series you should know
Several standard Maclaurin series are used again and again on the AP exam. Knowing them helps you build many other series.
1. Exponential
$$
$\displaystyle e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$
$$
2. Sine
$$
$\displaystyle \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$
$$
3. Cosine
$$
$\displaystyle \cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$
$$
4. Geometric series
$$
\displaystyle $\frac{1}{1-x}$=$\sum_{n=0}$^{$\infty$}x^n\quad \text{for } |x|<1
$$
This one is especially useful because many complicated rational functions can be rewritten using it.
5. Natural log form
$$
\displaystyle $\ln(1$+x)=$\sum_{n=1}$^{$\infty$}(-1)^{n+1}$\frac{x^n}{n}$\quad \text{for } -1<x\le 1
$$
6. Arctangent
$$
\displaystyle $\arctan$ x=$\sum_{n=0}$^{$\infty$}(-1)^n$\frac{x^{2n+1}}{2n+1}$\quad \text{for } |x|\le 1
$$
These formulas are not random memorization tricks. They act like building blocks. Once you know them, you can create many new series by changing the variable, multiplying, dividing, differentiating, or integrating.
How to find a Maclaurin series from a known series
A very common AP method is to start with a known series and transform it. For example, if you want the Maclaurin series for $e^{2x}$, substitute $2x$ into the known series for $e^x$:
$$
\displaystyle e^{2x}=$\sum_{n=0}$^{$\infty$}$\frac{(2x)^n}{n!}$=$\sum_{n=0}$^{$\infty$}$\frac{2^n x^n}{n!}$
$$
That is much faster than computing derivatives of $e^{2x}$ again and again.
Example: Find a Maclaurin series for $\sin(3x)$
Start with
$$
$\displaystyle \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$
$$
Replace $x$ with $3x$:
$$
$\displaystyle \sin(3x)=\sum_{n=0}^{\infty}(-1)^n\frac{(3x)^{2n+1}}{(2n+1)!}$
$$
You can also simplify it to
$$
$\displaystyle \sin(3x)=\sum_{n=0}^{\infty}(-1)^n\frac{3^{2n+1}x^{2n+1}}{(2n+1)!}$
$$
This is a valid Maclaurin series.
Example: Find a Maclaurin series for $\frac{1}{1+2x}$
Use the geometric series:
$$
$\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$
$$
Replace $x$ with $-2x$:
$$
\displaystyle $\frac{1}{1+2x}$=$\sum_{n=0}$^{$\infty$}(-2x)^n=$\sum_{n=0}$^{$\infty$}(-2)^n x^n
$$
This series converges when $|-2x|<1$, so the interval of convergence is
$$
$\displaystyle -\frac{1}{2}<x<\frac{1}{2}$
$$
Differentiating and integrating series to create new ones
Sometimes a function does not match a known series immediately, but its derivative or integral does. In those cases, you can work backward.
Differentiation example
Suppose you want a series for $\frac{1}{(1-x)^2}$. Start with
$$
$\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$
$$
Differentiate both sides:
$$
$\displaystyle \frac{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}$
$$
You may also rewrite it as
$$
$\displaystyle \frac{1}{(1-x)^2}=\sum_{n=0}^{\infty}(n+1)x^n$
$$
Differentiation changes the interval of convergence only by possibly losing an endpoint, so you should always recheck convergence after the manipulation.
Integration example
To find a series for $\ln(1+x)$, you can start with
$$
\displaystyle $\frac{1}{1+x}$=$\sum_{n=0}$^{$\infty$}(-1)^n x^n\quad \text{for } |x|<1
$$
Integrate both sides from $0$ to $x$:
$$
\displaystyle $\int_0$^x $\frac{1}{1+t}$\,dt=$\int_0$^x $\sum_{n=0}$^{$\infty$}(-1)^n t^n\,dt
$$
This gives
$$
$\displaystyle \ln(1+x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{n+1}}{n+1}$
$$
Reindexing produces the common form
$$
$\displaystyle \ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}$
$$
Finding a Taylor series centered at a value other than $0$
A Taylor series does not have to be centered at $0$. If the center is $a\neq 0$, the formula uses powers of $x-a$:
$$
$\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$
$$
This is useful when a function behaves nicely near a certain point. For example, if a problem asks for a series centered at $x=2$, your terms must use $(x-2)^n$, not x^n`.
Example idea
If you know the Maclaurin series for $\frac{1}{1-x}$, then to find a series for $\frac{1}{1-(x-2)}$ you can substitute $x-2$ for $x$ in the original series. That produces a series centered at $x=2$.
In AP Calculus BC, the key skill is recognizing the center and adjusting the variable accordingly. The center is not just a detail; it determines the form of the entire series.
How this fits into approximation and error đź“Ź
A major reason Taylor and Maclaurin series matter is that they approximate functions with polynomials. A polynomial partial sum like
$$
$\displaystyle P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$
$$
is called the $n$th Taylor polynomial. The more terms you use, the better the approximation usually becomes near the center $a$.
For example, the first few Maclaurin polynomials for $e^x$ are
$$
$\displaystyle P_0(x)=1$
$$
$$
$\displaystyle P_1(x)=1+x$
$$
$$
$\displaystyle P_2(x)=1+x+\frac{x^2}{2}$
$$
$$
$\displaystyle P_3(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}$
$$
These polynomials help estimate values like $e^{0.1}$ or $\sin(0.2)$ without a calculator’s built-in function. For small $x$, the approximation is often very accurate.
The error after using a Taylor polynomial is connected to the next unused term or the remainder. In many AP problems, the Lagrange error bound is used:
$$
$\displaystyle |R_n(x)|\le \frac{M|x-a|^{n+1}}{(n+1)!}$
$$
where $M$ is a bound on the next derivative over the interval. This helps you know how many terms are needed to achieve a desired accuracy.
Conclusion
students, finding Taylor or Maclaurin series is about turning complicated functions into useful infinite polynomial forms. The most efficient strategy is usually to start with a known basic series and transform it using substitution, algebra, differentiation, or integration. When the center is $0$, the series is Maclaurin; when the center is another value $a$, it is Taylor. These ideas are important because they connect function behavior, approximation, convergence, and error in one powerful toolset.
On the AP Calculus BC exam, being able to build a series correctly is just as important as recognizing whether the series converges. Mastering these methods gives you access to a major part of infinite sequences and series, and it prepares you to handle approximation problems with confidence ✨
Study Notes
- A Taylor series centered at $a$ is
$$
$ \displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$
$$
- A Maclaurin series is a Taylor series with $a=0$.
- Know the basic Maclaurin series for $e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$, $\ln(1+x)$, and $\arctan x$.
- Use substitution to create new series, such as replacing $x$ with $2x$ or $x-3$.
- Differentiate or integrate known series when the target function is related.
- Always check the interval of convergence after transforming a series.
- Taylor polynomials are finite approximations made from the first few terms of a Taylor series.
- Error bounds help measure how accurate the approximation is.
- The center matters: powers must match $x-a$ for a Taylor series centered at $a$.
- Series are powerful because they convert difficult functions into polynomial-like forms that are easier to analyze and compute.