Unit 10: Infinite Sequences and Series

Executive Summary

Series ask whether infinitely many terms can sum to a finite number. Convergence tests answer that question; Taylor series turn functions into polynomials.

Geometric series and p-series

Geometric: ∑arⁿ converges to a/(1-r) when |r| < 1; diverges otherwise. p-series: ∑1/nᵖ converges when p > 1 (harmonic series p = 1 diverges). These two are the reference points for every comparison.

Convergence tests

nth term: terms not approaching zero → divergence (the converse is false). Integral test: compare with an improper integral. Comparison tests: bound the terms by a known series (limit comparison uses the ratio of terms). Alternating series: terms decreasing to zero → convergence; the error of a partial sum is less than the next term. Ratio test: limit of |a(n+1)/a(n)| — less than 1 converges, greater than 1 diverges, equal to 1 inconclusive.

Absolute vs conditional convergence

Absolutely convergent: ∑|aₙ| converges (strongest). Conditionally convergent: ∑aₙ converges but ∑|aₙ| diverges — alternating series often land here. Absolute convergence implies convergence.

Taylor and Maclaurin series

Maclaurin series (centered at 0): eˣ, sin x, cos x, 1/(1-x) are the building blocks. Taylor series centered at c generalizes them. Substitution, differentiation, and integration generate new series from known ones. The Lagrange error bound measures how far a Taylor polynomial is from the function.

Power series

A power series converges on an interval (radius of convergence R). Test endpoints separately. The ratio test usually finds R.

Exam traps

The nth term test proves DIVERGENCE only. Ratio test = 1 proves nothing. The alternating series error bound uses the NEXT term. Check endpoints of the interval of convergence.