Convergence of Series Using Direct Comparison Test

Use the direct comparison test to determine if the series                                                             u000f n=1 }                                                               $n=1}^{        {   {  

When faced with the series given in the problem, the primary goal is to determine its convergence or divergence using the direct comparison test. This requires finding a second series that is either consistently larger or smaller than the given series and whose convergence properties are already known. Essentially, the direct comparison test is a valuable tool in determining the behavior of series by leveraging known series.

In this particular problem, the challenge lies in identifying or constructing a series that effectively bounds the given series above or below for all terms. The series rac{1}{4 + \ n}\\ decreases as $n$ increases, which suggests that it might be comparable to a well-known convergent or divergent series. A useful approach might be to simplify or manipulate the denominator to identify a simpler series with similar behavior, such as a p-series, which is a cornerstone in series comparison due to its well-defined convergence properties.

Conceptual understanding of the direct comparison test and its applications extends beyond this single series problem. It plays a critical role when dealing with complex series, where direct calculation or standard tests (like the p-series or geometric series tests) may not easily apply. Understanding how to dissect a series and ingeniously select a comparator series requires practice and deep familiarity with various types of series and their convergence behaviors.