Calculus 2

The root test decides whether a series converges by taking the nth root of the size of each term. Find L = lim ⁿ√|aₙ|. If L < 1, the series converges absolutely. If L > 1 (or L = ∞), it diverges. If L = 1, the test gives no answer.

The ratio test decides whether a series converges by looking at the ratio of consecutive terms. Find L = lim |aₙ₊₁/aₙ|. If L < 1, the series converges absolutely. If L > 1 (or L = ∞), it diverges. If L = 1, the test gives no answer.

A series converges absolutely if the series of absolute values converges. It converges conditionally if the series converges but the series of absolute values diverges. Absolute convergence always implies convergence. For example, the sum of (-1)^n/n² converges absolutely, while the sum of (-1)^n/n converges conditionally.

The alternating series test says that an alternating series with terms (-1)^(n+1) b_n, where b_n > 0, converges if two conditions hold: the sizes b_n decrease (at least from some point on), and b_n approaches 0. For example, the alternating harmonic series 1 – 1/2 + 1/3 – 1/4 + … converges.

The limit comparison test says that if a_n > 0 and b_n > 0 and the limit of a_n/b_n as n goes to infinity is c, where c is a positive finite number, then the sum of a_n and the sum of b_n either both converge or both diverge. You pick b_n as a simple series that behaves like a_n for large n.

The direct comparison test compares a series with positive terms to a series you already know, term by term. If 0 ≤ a_n ≤ b_n and the sum of b_n converges, then the sum of a_n converges. If a_n ≥ b_n ≥ 0 and the sum of b_n diverges, then the sum of a_n diverges.

A p-series is a series of the form 1/1^p + 1/2^p + 1/3^p + …, where p is a constant. The p-series test says it converges when p > 1 and diverges when p ≤ 1. For example, the sum of 1/n² converges, while the harmonic series (p = 1) and the sum of 1/√n diverge.

The integral test says that if a_n = f(n), where f is positive, continuous and decreasing for x ≥ 1, then the series of a_n and the improper integral of f(x) from 1 to infinity either both converge or both diverge. The integral decides convergence, but its value is not the sum.

A telescoping series is a series whose terms can be written as differences, a_n = b_n – b_(n+1), so that most of each partial sum cancels. The nth partial sum collapses to S_n = b_1 – b_(n+1), and the sum of the series is b_1 minus the limit of b_(n+1), if that limit exists.

The divergence test, also called the nth term test, says that if the terms of a series do not approach 0, the series diverges. In symbols, if the limit of a_n as n goes to infinity is not 0 or does not exist, then the sum of a_n diverges. If the limit is 0, the test tells you nothing.