Power Series

A series of the form:

Here:

• - a variable
• - a constant

Convergence of a power series can be checked using ratio test or root test.

Radius of convergence

Maximum radius of in where the series converges.

The below equation can be used to find :

The series may converge or diverge for .

Range of convergence

is the range of convergence. Aka. interval of convergence. The series may converge or diverge at the endpoints. Endpoints must be checked separately to find out if they must be included in the range of convergence.

The series is:

• Absolutely converging for
• Diverging for
• Uniformly converging for

Theorem 1

Suppose and . Then ( is uniformly (and absolutely) converging.

Proof Hint

• Note the relation between and
• Prove is an upperbound to , using it’s infinity limit
• Define
• Prove is a bound to
• Prove is converging as