Series

Series and Its Convergence

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

Suppose and are series with ultimately non-negative terms. If ​ for all sufficiently large , and ​ converges, then ​ also converges. Conversely, if for all sufficiently large , and diverges, then also diverges.

There is also a limit version of the 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 .

Power Series

Radius of Convergence

Let be a power series that converges absolutely for and diverges for . Such an is called the radius of convergence of the power series.

Calculating The Radius

For the power series , the radius of convergence is given by

Proof The first formula comes from the root test, and the second from the ratio test.

Remark

Power series has nicer properties in the complex case. In fact, this formula of the radius of convergence can be extended to the complex power series. (See theorem)