Direct Comparison Test for Series Convergence

Use the direct comparison test to determine if the series $\sum_{n=1}^{\infty} \frac{2^n}{7^n + 8}$ converges or diverges.

When determining the convergence or divergence of an infinite series, one useful method is the direct comparison test. This technique involves comparing the given series to another series whose behavior (convergence or divergence) is already known. If a given series has positive terms that are each less than or equal to the terms of a second convergent series, then the original series also converges. Conversely, if the terms of the given series are greater than or equal to those of a divergent series, then the series diverges.

For this problem, you're asked to analyze the series represented by the sum of two raised to the nth power over the quantity seven raised to the nth power plus eight, from n equals one to infinity. This series is interesting as it resembles a geometric series, which often serves as a good point of comparison. You can leverage the structure of this series to compare it against a geometric series with a simpler form, possibly like two-sevenths raised to the nth power.

Understanding the direct comparison test requires a good grasp of inequalities and the ability to conceptually visualize the growth rates of series. This problem is a classic example found within comparison tests, a key topic in the study of series within calculus courses.