Power Series

Definition

1. Let $a\in \mathbb{R}$
2. The power series about $a$ denoted as $\{C_n{\}}_{n=0}^{\infty }\subset \mathbb{R}$ is a Series of Real Numbers of the form $C_0+C_1(x-a)+C_2(x-a{)}^2+\dots$ Where:

• $C_n$ is called the $n-th$ term coefficient of the PS
• $a$ is the center of the PS

Convergence/Divergence

• Test with Ratio Test

Theorems

• Power Series Convergence Possibilities Theorem

Examples

Example 1

${\sum }_{n=1}^{\infty }\frac{(x-3{)}^n}{4^n}$ is a power series where:

• $C_n=\frac{1}{4^n}$
• $a=3$

Example 2

${\sum }_{n=2}^{\infty }\frac{(4x+1{)}^n}{2^n\sqrt{n}}$ is a power series where:

• $a=\frac{1}{4}$
• $C_n=\frac{2^n}{\sqrt{n}}$