Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read · Updated 2026-05-10
Finding Taylor or Maclaurin series for a function — AP Calculus BC
AP Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read
1. Core Definitions of Taylor and Maclaurin Series★★☆☆☆⏱ 2 min
Finding a Taylor or Maclaurin series means constructing an infinite power series representation of a given infinitely differentiable function. Coefficients are defined to match the function's value and all orders of derivative at a chosen center point $a$.
This topic is a core skill in Unit 10, which makes up 17-18% of the AP Calculus BC exam. Questions on finding series appear in both multiple-choice and free-response sections, and are often combined with other series skills like finding convergence or approximation.
2. Direct Computation via the Definition Method★★★☆☆⏱ 3 min
The formal definition of a Taylor series centered at $a$ gives the general formula:
Where $f^{(n)}(a)$ is the $n$-th derivative of $f$ evaluated at $a$, and $f^{(0)}(a) = f(a)$ by convention. For Maclaurin series ($a=0$), this simplifies to:
The direct method requires computing the first few derivatives, identifying a pattern in coefficients, and writing the general term. It is most often used for functions that do not fit a known standard series, or for non-zero centers.
Exam tip: Use this method when the question explicitly asks you to use the definition of Taylor series, or for non-zero centers.
3. Constructing Series by Substitution★★★☆☆⏱ 3 min
Once you memorize standard Maclaurin series for common functions, you do not need to recompute coefficients from scratch for most exam problems. Substitution is the most common shortcut: if you have a known series $f(u) = \sum_{n=0}^{\infty} c_n u^n$, then $f(g(x)) = \sum_{n=0}^{\infty} c_n (g(x))^n$ within the new interval of convergence.
This method is tested more frequently than the direct definition method on the AP exam, because it tests your ability to manipulate known series rather than just compute derivatives.
4. Constructing Series by Differentiation and Integration★★★★☆⏱ 3 min
Power series can be differentiated and integrated term-by-term within their interval of convergence, so you can use this property to derive a new series from a known one. If $f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$, then:
f'(x) = \sum_{n=1}^{\infty} n c_n (x-a)^{n-1}
\int f(x) dx = C + \sum_{n=0}^{\infty} \frac{c_n}{n+1} (x-a)^{n+1}
Term-by-term differentiation and integration do not change the radius of convergence of the original series; only convergence at the endpoints may change. This method is used when your target function is the derivative or integral of a function with a known series.
5. Adjusting Series by Multiplying by a Power of $x$★★★☆☆⏱ 3 min
A common AP exam problem requires finding the series for a function that is a known function multiplied by $x^k$ (or $(x-a)^k$ for a general center). This simple manipulation checks your understanding of how power series terms work: when you multiply a power series by $(x-a)^k$, every exponent increases by $k$, and coefficients do not change.
Common Pitfalls
Why: Students forget to apply the exponent to the negative sign, only applying it to the $x^2$ term.
Why: Students confuse the general form of a Taylor series with pre-memorized Maclaurin coefficients, assuming shifting $x$ is enough for a non-zero center.
Why: Students forget that $\ln(1+0) = 0$, so the constant of integration evaluated at the center is zero.
Why: Students get used to derivatives of $e^x$ and forget the chain rule gives an extra constant factor for the linear inner function.
Why: Students forget that multiplying by $x^2$ shifts all exponents up by 2.