FoldUnfold Table of Contents Determining Power Series Representations of Rational Functions Example 1 Example 2 Example 3 Example 4 Determining Power Series Representations of Rational Functions We are now going to look at some examples of determining power series representations for rational functions. Before we do so, we must recall a very important power series […]

## Cauchy Product of Power Series

FoldUnfold Table of Contents Cauchy Product of Power Series Example 1 Example 2 Cauchy Product of Power Series The following theorem will give us a way to in a sense, “multiply” two power series together. Theorem 1 (The Cauchy Product of Power Series): Consider the power series $\sum_{n=0}^{\infty} a_nx^n$ with a radius of convergence $R_1$, […]

## Algebraic Operations of Power Series Examples 1

FoldUnfold Table of Contents Algebraic Operations of Power Series Examples 1 Example 1 Example 2 Algebraic Operations of Power Series Examples 1 Recall from the Algebraic Operations of Power Series page that: If the power series $\sum_{n=0}^{\infty} a_n x^n$ has radius of convergence $R_1$ and if $k \in \mathbb{R}$ is some constant, then the power […]

## Algebraic Operations of Power Series

FoldUnfold Table of Contents Algebraic Operations of Power Series Example 1 Example 2 Algebraic Operations of Power Series We will now begin to look at some algebraic properties of power series. Theorem 1: Consider the power series $\sum_{n=0}^{\infty} a_nx^n$ with a radius of convergence $R_1$, and let $k$ be some constant. Then the power series […]

## The Radius of Convergence of a Power Series Examples 3

FoldUnfold Table of Contents The Radius of Convergence of a Power Series Examples 3 Example 1 Example 2 The Radius of Convergence of a Power Series Examples 3 Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series $\sum_{n=0}^{\infty} a_n(x – c)^n$ […]

## The Radius of Convergence of a Power Series Examples 2

FoldUnfold Table of Contents The Radius of Convergence of a Power Series Examples 2 Example 1 Example 2 Example 3 The Radius of Convergence of a Power Series Examples 2 Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series using the […]

## The Radius of Convergence of a Power Series Examples 1

FoldUnfold Table of Contents The Radius of Convergence of a Power Series Examples 1 Example 1 Example 2 Example 3 The Radius of Convergence of a Power Series Examples 1 Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series using the […]

## The Radius of Convergence of a Power Series

FoldUnfold Table of Contents The Radius of Convergence of a Power Series Determining the Radius of Convergence of a Power Series Example 1 Example 2 The Radius of Convergence of a Power Series Recall from the Power Series page that we saw that a power series will converge at it’s center of convergence $c$, and […]

## Power Series

FoldUnfold Table of Contents Power Series Power Series We will now look at an important type of series known as a power series which we define as follows. Definition: A Power Series in Powers of $x – c$ is a series that takes on the form $\sum_{n=0}^{\infty} a_n(x – c)^n = a_0 + a_1(x – […]

## Rearrangement of Terms in Convergent Series

FoldUnfold Table of Contents Rearrangement of Terms in Convergent Series Rearrangement of Terms in Convergent Series Thus far we have looked at a series that converges to have only one sum. We will now note that if a series is conditionally convergent, then rearranging the terms in this series will allow us to change the […]

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