Radius of Convergence of a Power Series with Factorial Coefficients

Determine the radius of convergence for the power series with $c_n = n!$ and $(x - 2)^n$.

When solving for the radius of convergence of a power series, it's essential to analyze the general form of the series in order to apply an appropriate convergence test. In this problem, the coefficients of the series are given by n-factorial, denoted as n!, and the series is centered around x = 2. To determine the convergence of such series, the Ratio Test or Root Test are commonly employed due to the factorial nature of the coefficients, which tend to grow quite rapidly.

The Ratio Test is particularly useful in this scenario. It involves taking the limit of the ratio of successive terms in the series as n approaches infinity. The factorial coefficients are significant here as they involve rapid growth, which affects the convergence characteristics. By setting up the ratio of $(n+1)!$ to $n!$ and investigating the limit, one can determine whether the series converges absolutely within a certain distance from the center of the series. This distance from the center is known as the radius of convergence.

Exploring power series with varying coefficients such as these factorials not only provides insight into convergence behavior but also enhances the understanding of analytic properties of functions represented by power series. Understanding how to manipulate and analyze these series is a crucial skill in mathematical analysis, particularly in topics related to function representation and complex analysis.