AP Calculus BC - Unit 9 Convergence Tests

Geometric Test

If it is a Geometric Pattern with an r < 1 then the series converges. a / 1 - r where a is the first term. Make sure it is in the format of a(r)^n

Telescoping Test

A series converges if the partial sums collapse to a simple expression after canceling out terms. It helps identify convergence/divergence.

Integral Test

A method in calculus to determine if an infinite series converges or diverges by comparing it to an improper integral. Must be positive, continuous, and decreasing. Both either converge or diverge.

P-Series pattern

The P-Series pattern is a series of the form ∑(1/n^p), where p is a positive constant. It converges if p > 1 and diverges if p ≤ 1.

Comparison Test

Comparison Test: If 0 ≤ aₙ ≤ bₙ for all n and ∑bₙ converges, then ∑aₙ also converges. If ∑bₙ diverges, then ∑aₙ also diverges.

Alternating Series Test

The Alternating Series Test states that an alternating series converges if the terms decrease in absolute value and approach zero.

Ratio Test

Ratio Test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges.

Root Test

The Root Test is a mathematical test used to determine the convergence of an infinite series by analyzing the limit of the nth root of the absolute value of the terms in the series.