Series

Series Convergence

Let and . We say that the terms are the partial sums of the series . We say the series converges if exists, and diverges otherwise.

e.g.

• Geometric series converges to for and diverges otherwise.
• Harmonic series diverges.

Absolute Convergence

A series converges absolutely if converges. A series converges conditionally if it converges, but not absolutely.

Tests for Convergence

Integral Test

Suppose where is a positive non-increasing integrable function on . Then converges if and only if the improper integral converges.

e.g. The -series converges if and only if .

Simple Comparison Test

Limit Comparison Test

Suppose and are series with ultimately non-negative terms. If then converging implies converges. If then diverging implies diverges.

e.g. Consider . Let . Then . Since converges, so does .

Ratio Test

Suppose is a series and let . Then converges absolutely if , diverges if , undetermined if .

Radius of Convergence

Let be a power series. The radius of convergence is defined as . The series converges absolutely for and diverges for . The behaviour at the endpoints must be checked separately.