Convergence of Alternating Series with Rational Function Term

Consider the series which goes from 1 to infinity: $(-1)^{n+1} \cdot \frac{5n + 3}{2n - 7}$. Will this series converge or diverge?

This problem deals with evaluating whether a given infinite series converges or diverges. We are specifically dealing with an alternating series, indicated by the factor of negative one raised to the power of n plus one. Alternating series introduce unique characteristics compared to regular series because their terms switch between positive and negative. A crucial tool in analyzing such series is the Alternating Series Test, which allows us to determine convergence if two conditions are satisfied: the absolute value of the terms decreases monotonically to zero, and the limit of the terms as n approaches infinity is zero. In this context, the terms of the series are dictated by a rational function of n, and understanding how this function behaves is key to applying the test effectively.

Moreover, even if an alternating series converges according to the Alternating Series Test, it's valuable to investigate whether it converges absolutely or conditionally. Absolute convergence occurs if the series of absolute values also converges, which often requires comparison tests. Conditional convergence happens when the series converges, but the series of absolute values diverges. This distinction plays a significant role in understanding the kind of convergence that is occurring and can affect the sum's approximation particularly when truncating the series after a finite number of terms. Understanding these concepts not only helps in determining convergence but also deepens comprehension of series behavior and their practical implications in calculus.