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AP CALCULUS BC · MODULE 07

Power Series & Taylor/Maclaurin Series
Polynomial Approximations of Functions

Learn to represent functions as infinite polynomials — power series, radius of convergence, Taylor polynomials, error bounds, and series manipulations.

📘 BUders Learning System
This module covers power series, Taylor and Maclaurin series, radius of convergence, Taylor polynomials, Lagrange error bound, and series manipulations. All topics are BC Calculus only.
⚡ 01
Power Series
$\sum_{n=0}^\infty c_n (x-a)^n$ — definition, interval and radius of convergence, ratio/root tests.
🔷 BC only → start
📐 02
Taylor & Maclaurin Series
$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ — Maclaurin series when $a=0$.
🔷 BC only → start
📋 03
Common Maclaurin Series
$e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$, $\ln(1+x)$, $\arctan x$ — memorized series.
🔷 BC only → start
📊 04
Taylor Polynomials & Approximations
$P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k$ — approximating functions with polynomials.
🔷 BC only → start
🎯 05
Lagrange Error Bound
$|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$ — error estimation for Taylor polynomials.
🔷 BC only → start
🔧 06
Series Manipulations
Substitution, differentiation, integration of power series — creating new series from known ones.
🔷 BC only → start
📝 07
Practice Problems
20 solved problems covering all power series and Taylor series topics.
practice → start
🧪 08
Test Yourself
20-question interactive quiz with instant feedback.
assessment → start