Interval of Convergence Using Ratio Test

Using the ratio test, determine the intervals of convergence for the power series 28 29. Answer: The interval of convergence is $2 < x < 4$.

When tackling the problem of finding the interval of convergence for a power series, the ratio test is a powerful tool. The fundamental concept of the ratio test is to examine the limit of the ratio of consecutive terms in the series. If this limit is less than one, the series converges absolutely. For a power series centered at some point $a$, understanding the behavior of this ratio helps determine where the series converges on the real line.

The focus of this problem involves analyzing a series of the form 28 29. The step requiring the most attention is forming the ratio of consecutive terms and simplifying. Solving the inequality derived from the ratio test reveals the values of $x$ for which the series converges.

Recognizing patterns and symmetries in the series simplifies the work. Additionally, verifying endpoints separately is crucial when it comes to determining the complete interval of convergence. In terms of the underlying concepts, this kind of problem reinforces understanding of series and convergence, and specifically applies the ratio test as a method for evaluating the behavior of power series around their center of convergence.