Limit Comparison Test for Series Convergence23

Determine if the series $\frac{1}{3^n + 5}$ converges or diverges using the limit comparison test.

In the study of infinite series, determining whether a series converges or diverges is a fundamental concept, especially in calculus and analysis courses. The limit comparison test is a powerful tool used to determine the behavior of a series by comparing it to another series whose convergence properties are already known. This approach is particularly useful when dealing with complex expressions that do not easily lend themselves to more straightforward convergence tests, like the p-series test or geometric series test.

The limit comparison test states that for two series with positive terms, sum a_n and sum b_n, if the limit of a_n/b_n as n approaches infinity is a positive finite number, both series either converge or diverge together. This method relies on the intuitive idea that if two sequences behave similarly for large values of n, then their series should behave similarly in terms of convergence. This requires selecting a b_n that is simple enough to allow the limit to be evaluated but still reflects the dominant behavior of a_n.

In this particular problem, the series given by 1/(3ⁿ + 5) is under analysis. By selecting an appropriate comparative series, often a geometric or p-series, you can exploit the properties of the more familiar series to establish convergence or divergence. This approach highlights the strategic aspect of problem-solving in mathematics, where identifying symmetries or analogous situations allows existent theoretical results to be applied effectively, simplifying the problem-solving process.