Taylor & Maclaurin Series: Complete AP® Calculus BC Guide
Master power series expansions, essential Maclaurin series formulas, interval of convergence testing, and the Lagrange Error Bound for high-scoring FRQs.
1. Taylor & Maclaurin Polynomial Formulation
A Taylor polynomial approximates a smooth differentiable function $f(x)$ near a specified center $x = c$. The $n$th-degree Taylor polynomial is defined by matching derivatives of $f$ at $c$:
When the center is $c = 0$, the series is known as a Maclaurin Polynomial. Memorizing coefficient relationships saves immense time on Free Response Questions. For derivative calculations, consult our derivative rules guide.
2. Must-Memorize Maclaurin Series ($e^x, \sin x, \cos x, \frac{1}{1-x}$)
The College Board expects every AP® Calculus BC student to know the following foundational Maclaurin series and their intervals of convergence by heart:
| Function | Series Expansion | General $n$th Term | Interval of Convergence |
|---|---|---|---|
| $\frac{1}{1 - x}$ | $1 + x + x^2 + x^3 + \dots$ | $x^n$ | $(-1, 1)$ |
| $e^x$ | $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ | $\frac{x^n}{n!}$ | $(-\infty, \infty)$ |
| $\sin x$ | $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$ | $(-1)^n \frac{x^{2n+1}}{(2n+1)!}$ | $(-\infty, \infty)$ |
| $\cos x$ | $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$ | $(-1)^n \frac{x^{2n}}{(2n)!}$ | $(-\infty, \infty)$ |
| $\ln(1 + x)$ | $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$ | $(-1)^{n+1} \frac{x^n}{n}$ | $(-1, 1]$ |
3. Algebraic Substitution, Differentiation & Integration of Series
Rather than building new series from scratch using repeated derivatives, the most efficient approach is manipulating known series through algebraic substitution, multiplication, term-by-term differentiation, or term-by-term integration:
- Substitution: To find the Maclaurin series for $\cos(3x^2)$, substitute $u = 3x^2$ into the known cosine series:
$$\cos(3x^2) = 1 - \frac{(3x^2)^2}{2!} + \frac{(3x^2)^4}{4!} - \dots = 1 - \frac{9x^4}{2} + \frac{81x^8}{24} - \dots$$
- Integration: To integrate non-elementary functions like $\int e^{-x^2}\,dx$, replace $e^{-x^2}$ with its series and integrate term by term. Check integration details in our integral rules reference.
4. Lagrange Error Bound (Taylor's Remainder Theorem)
When approximating $f(x)$ with an $n$th-degree Taylor polynomial $P_n(x)$ centered at $c$, the absolute truncation error $|R_n(x)| = |f(x) - P_n(x)|$ is bounded by the Lagrange Error Bound formula:
Lagrange Error Bound Formula
Where $M$ is the maximum value of $|f^{(n+1)}(t)|$ on the closed interval between $c$ and $x$.
$$|R_3(2.3)| \le \frac{12}{(3 + 1)!} |2.3 - 2|^4 = \frac{12}{24} (0.3)^4 = 0.5 \times 0.0081 = 0.00405$$ Therefore, the approximation is guaranteed to be accurate within $\pm 0.00405$.
Review additional exam tricks on the AP Calculus review sheet.
Frequently Asked Questions (AP® Calculus AB & BC)
Q: How do I find f^(n)(0) from a given Maclaurin series?
Equate the coefficient of x^n in the given series to f^(n)(0) / n!. Therefore, f^(n)(0) = (coefficient of x^n) * n!. This is tested nearly every year on Section I of the BC exam.
Q: What is the difference between Lagrange Error Bound and AST Error Bound?
The Alternating Series Error Bound can ONLY be used if the series alternates in sign with terms decreasing to 0 (and the error is bounded by the next term). The Lagrange Error Bound works for ALL Taylor polynomials regardless of whether the series alternates.
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Why This Step? Blueprint Breakdown: AP® Calculus AB/BC
Key Conceptual Scaffolding & Diagnostic Intuition
Tested Competency: Derivative theorems, Riemann sums & Taylor series. In modern competitive exams, questions are engineered to test whether you understand the underlying structural relationship—not just whether you memorized a formula.
Forgetting the chain rule on composite functions or mixing up Average Rate of Change with Instantaneous Rate of Change.
Always write out your hypotheses and state conditions explicitly before calculating test statistics. Graders award up to 40% of partial credit for proper setup, regardless of arithmetic precision.