The capstone of BC calculus. Decide whether infinite series converge using a toolbox of tests, then represent functions as Taylor, Maclaurin, and power series — bounding the error of each approximation. This is a BC-only unit and the largest on the exam.
All 15 topics from the College Board CED, in order. This entire unit is BC only.
Topic 10.1
Convergent & Divergent Series
Defining convergence via the sequence of partial sums.
Topic 10.2
Geometric Series
Sum a/(1 − r) when |r| < 1.
Topic 10.3
nth Term Test for Divergence
If terms don’t → 0, the series diverges.
Topic 10.4
Integral Test
Comparing a series to an improper integral.
Topic 10.5
Harmonic & p-Series
Σ1/nᵖ converges iff p > 1.
Topic 10.6
Comparison Tests
Direct and limit comparison for convergence.
Topic 10.7
Alternating Series Test
Convergence for alternating, decreasing terms → 0.
Topic 10.8
Ratio Test
Using the limit of |aₙ₊₁/aₙ|.
Topic 10.9
Absolute vs. Conditional Convergence
Distinguishing the two kinds of convergence.
Topic 10.10
Alternating Series Error Bound
Bounding the error by the first omitted term.
Topic 10.11
Taylor Polynomial Approximations
Approximating a function near a point.
Topic 10.12
Lagrange Error Bound
Bounding Taylor-approximation error.
Topic 10.13
Radius & Interval of Convergence
Where a power series converges.
Topic 10.14
Taylor & Maclaurin Series
Building the full series of a function.
Topic 10.15
Functions as Power Series
Manipulating known series to represent functions.
About Unit 10
Unit 10 is the capstone of BC calculus. It begins with convergence: an infinite series Σaₙ converges when its sequence of partial sums approaches a finite limit. You build a toolbox of tests — the nth term test for divergence, geometric and p-series facts, the integral, comparison, alternating series, and ratio tests — and learn to tell absolute from conditional convergence.
The second half represents functions as series. A Taylor polynomial approximates a function near a point, with the Lagrange error bound (and the alternating series error bound) controlling accuracy. You find the radius and interval of convergence of a power series, construct Taylor and Maclaurin series for functions like eˣ, sin x, and 1/(1−x), and manipulate known series — differentiating, integrating, and substituting — to represent new functions.
On the exam this BC-only unit is 17–18% of the BC exam — the largest single unit — and takes about 34 class periods (BC). The four mathematical practices below run through every topic: