Infinite Sequences and Series Overview — AP Calculus BC
1. Unit at a Glance
We start with foundational definitions of convergence and divergence for infinite series, then build up a full toolkit of common convergence tests you will use repeatedly throughout the unit. After mastering convergence tests for numeric series, we move to power series, Taylor and Maclaurin polynomials and series, and methods to bound approximation error.
The logical flow builds from basic definitions to more complex applications, so it is important to master earlier topics before moving to more advanced concepts like error bounds and Taylor series. Each new test builds on previous knowledge to expand your ability to classify any series you encounter on the exam.
Common Pitfalls
Why: The nth term test only proves divergence when terms do not approach 0; it cannot confirm convergence.
Why: The ratio test only gives the radius of convergence; endpoints can still converge or diverge.
Why: A Taylor polynomial is a finite approximation, while a Taylor series is an infinite representation of a function.