Power series are a special kind of mathematical expression that can help us understand different functions. How these series come together, or "converge," depends on their kind, which changes how far they can reach, called the radius and interval of convergence. There are two main types of power series: Taylor series and Maclaurin series. Each has its own features that impact how they converge.

Radius of Convergence: The radius of convergence, often noted as ( R ), can be figured out using two tests: the Ratio Test or the Root Test. For a power series that looks like this:

$\sum_{n=0}^{\infty} a_n (x - c)^n,$

we find the radius using:

$R = \lim_{n \to \infty} \frac{|a_n|}{|a_{n+1}|},$

or

$R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}.$

Interval of Convergence: The interval of convergence shows the range where the power series works. It stretches from ( (c - R, c + R) ). However, we need to check the endpoints, or edges, at ( x = c - R ) and ( x = c + R ) to see if the series converges there too. How the series behaves at these points can change based on the function it represents.

Types of Power Series:

1. Taylor Series: A Taylor series is centered at a specific point and converges within a certain radius, heavily influenced by the function’s derivatives (slopes) at that point.
2. Maclaurin Series: This is a special kind of Taylor series that is centered at zero. It usually has a smaller radius of convergence. For example, functions like ( \frac{1}{1-x} ) converge more quickly than their Taylor series counterparts.

In summary, how a power series's coefficients and its center interact determines where it can effectively converge. This shows why it’s important to study this area in calculus.