Using the Root Test for Series Convergence

Use the root test to determine if the series $\left(\frac{3+5n}{2+3n}\right)^n$ converges, diverges, or is inconclusive.

The problem at hand asks us to apply the root test to the given series expression, a common technique when analyzing the convergence of series with a general term raised to the nth power. The root test provides a method of examining the asymptotic behavior of the nth root of sequence terms, offering insight into the series' convergence through comparisons to convergent geometric series. By considering the nth root of the absolute value of the general term, we focus on how quickly the terms approach zero as n approaches infinity. This behavior is crucial since it reveals whether the entire series will sum to a finite number or diverge. Generally, the root test is especially useful when the series terms contain powers of n, providing a tractable means of evaluating convergence by observing the series' growth rate. In this problem, recognizing the series' structure involves the simplification of the expression within the root and checking its limit as n grows large. Notably, the root test leads us to compare this limit to 1, which is a standard threshold in determining convergence. If this limit is less than 1, the series converges; if greater than 1, it diverges. If equal to 1, the test is inconclusive, necessitating alternative methods. Understanding this test and its outcomes are fundamental skills in calculus, particularly in courses focusing on series and convergence tests. Students dealing with such problems enhance their ability to identify appropriate tests for various series, laying a foundational understanding for more advanced mathematical studies.