Convergence of Series with Rational Function

Determine if the series $\sum_{n=1}^{\infty} \frac{n+1}{3n+2}$ converges or diverges, and justify your answer.

In this problem, we are asked to determine the convergence or divergence of a series. The series in question has terms of a rational function form, specifically $(n+1)/(3n+2)$. One of the initial approaches to consider is the Comparison Test, as it allows us to compare the given series with a simpler series that we already know converges or diverges. For instance, the given series resembles a harmonic series, as the terms simplify in a straightforward manner, possibly leading to comparison with a known p-series.

Another approach is the Limit Comparison Test. This method is particularly useful when the series in question is a rational function. By dividing the terms appropriately and analyzing the limit of the resulting expression, we can contrast the behavior of our series with that of a simpler series, often leading to a definitive conclusion about convergence.

Overall, understanding series and their behavior is crucial in various fields of mathematics, including calculus and analysis. This problem, in particular, emphasizes the importance of recognizing series forms and propels students towards deeper analytical techniques required for solving convergence problems. Mastery of tools like the Comparison and Limit Comparison Tests forms the foundation for tackling more complex series seen in advanced mathematics.