A power series is any infinite series of the following form:

Where is some sequence and is some constant called the centre of the series, which is usually equal to .

Radius of Convergence

Given the power series of the form the radius of converge is defined as follows:

There are 3 main cases for the radius of convergence depending on the result.

Cases

1. There is a Real Number

• If , the series converges
• If , the series diverges

In both these instances it is said to have a radius of convergence of .

2. The Series Converges for All Values of

and the series is said to have an infinite radius of convergence.

3. The Series Diverges for All Values of

and the series is said to have a radius of convergence of .

Interval of Convergence

The interval of convergence is the set of values for which the power series converges.

Operations

Given two power series and the following operations can be performed.

Addition

When two power series are added the resulting series has a radius of convergence greater than or equal to the smallest radius of convergence of the terms.

For example if has a radius of convergence of and has a radius of convergence of then must have a radius of convergence of at least .

Multiplication by a Constant

Multiplication by a Polynomial

Differentiation

Notice how because the first term is just equal to which is a constant it can be removed.

Hence the radius of convergence stays the same.

Integration