Convergence of Series Using the Root Test

Use the root test on the series $\left(\frac{n^n}{2^{1+4n}}\right)$ to determine its convergence or divergence.

The root test is a powerful tool used to determine the convergence or divergence of infinite series. It involves taking the n-th root of the absolute value of the terms in the sequence and analyzing the limit as n approaches infinity. This test is particularly useful for series where terms involve powers that become complex to manage using other tests. When applying the root test, if the limit is less than one, the series converges absolutely. If it is greater than one, the series diverges. If the limit equals one, the test is inconclusive, and other methods must be used to determine convergence.

In this problem, you are dealing with a series that includes exponential terms, typical when using the root test. The key to solving this problem is simplifying the expression inside the series. Pay special attention to the manipulation of powers and bases since proper simplification can make the application of the root test straightforward. Moreover, review the properties of exponents and how they interact with limits because a firm grasp of these concepts will guide you to the correct conclusion more easily.

The root test often overlaps in utility with the ratio test. Understanding both can provide deeper insights into the nature of series. In some cases, trying both tests may offer a clearer pathway to arriving at a conclusion. As with any test for convergence, practice with a range of series is vital in developing intuition for selecting the most appropriate test and correctly interpreting its results.