Interval of Convergence for Power Series with Factorial Coefficients

Find the interval of convergence for the power series with $c_n = \frac{1}{n!}$ and $(x - 2)^n$.

When dealing with power series, an important concept is the interval of convergence, which tells us the set of values for which the series converges. To find the interval of convergence, one typically applies convergence tests such as the Ratio Test or the Root Test, depending on the series structure. In this problem, we are given a power series with factorial coefficients in the form of one over n factorial. These types of coefficients often lend themselves well to the Ratio Test, largely due to how they behave when dividing successive terms. Factorials grow very rapidly, which typically influences the series to have a finite radius of convergence.

Understanding the nature of the factorial within the power series is crucial. The Ratio Test helps determine the radius of convergence by analyzing the limit of the ratio of consecutive terms. This test is particularly effective here because the factorial function in the denominator simplifies well under division, rendering the calculation of the limit straightforward. After determining the radius, the convergence at the endpoints needs to be checked separately by substituting them back into the original series and testing for convergence of the resulting simpler series.

In this type of problem, recognizing patterns in series behavior can significantly reduce the complexity of finding the solution. Furthermore, understanding the general behavior of elementary functions, such as exponential functions whose Maclaurin series include factorials, can help in forming intuition about why certain series converge and others do not. Thus, identifying series convergence is not just about computation but also about developing an intuition based on mathematical principles and previous experience with similar problems.