Unit 10: Infinite Sequences and Series (BC)
Work through these 15 topics in order. Clear the gatekeeper and the rest of the course opens up.
10.1 Convergent and divergent series (BC)
An infinite series is the limit of its partial sums.
10.2 Geometric series (BC)
A geometric series multiplies by the same ratio each term.
10.3 The nth-term test for divergence (BC)
If the terms themselves do not shrink to zero, there is no way the running total can settle — the series must diverge.
10.4 The integral test (BC)
If the terms come from a positive, decreasing function, the series and the matching improper integral are like a staircase and a ramp — they rise together, so they converge or diverge together.
10.5 Harmonic series and p-series (BC)
A p-series \sum 1/n^p converges exactly when the terms shrink fast enough — that threshold is p>1.
10.6 Comparison and limit-comparison tests (BC)
Compare a messy positive series to a known one (usually a p-series or geometric).
10.7 The alternating series test (BC)
When terms alternate sign and steadily shrink to zero, the partial sums zig-zag inward toward a limit — like closing in from both sides.
10.8 The ratio test (BC)
Look at how each term compares to the one before.
10.9 Absolute vs conditional convergence (BC)
A series can converge only because of sign cancellation (conditional) or robustly even if you make every term positive (absolute).
10.10 The alternating series error bound (BC)
Because an alternating series’ partial sums straddle the true sum, the error after stopping is no bigger than the first term you left out.
10.11 Taylor polynomial approximations (BC)
A Taylor polynomial matches a function’s value and first several derivatives at a center, so it hugs the curve near that point — the more terms, the closer the fit.
10.12 The Lagrange error bound (BC)
The error of a Taylor approximation is controlled by the size of the next derivative over the interval — it behaves like the first term you dropped.
10.13 Radius and interval of convergence (BC)
A power series converges only within a certain distance of its center — the radius.
10.14 Taylor and Maclaurin series (BC)
Letting a Taylor polynomial run to infinitely many terms gives a power series that equals the function on its interval of convergence.
10.15 Representing functions as power series (BC)
New series come cheaply from known ones: substitute, multiply, differentiate, or integrate a known series term by term to represent a related function.