Integral Test for Series Convergence

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

The integral test is a fundamental tool in calculus for analyzing the convergence of series. It provides a bridge between discrete series and continuous functions, allowing one to assess the behavior of an infinite series by examining the integral of its associated function. Specifically, it applies to series with positive, decreasing terms. By considering the integral of the corresponding function, one can determine whether the series converges or diverges. This method is particularly useful when direct convergence tests are challenging to apply.

In this problem, we are tasked with using the integral test to decide the convergence of the series where the general term is n divided by the sum of n squared and one. The integral test involves integrating the function $f(x) = \frac{x}{x^2 + 1}$ from 1 to infinity. A key aspect here is recognizing when the integral diverges or converges. Understanding the behavior of the function and evaluating this improper integral will lead to insights about the series.

This problem not only involves applying the integral test but also requires a strong grasp of integration techniques for rational functions. Grasping such concepts provides a deeper understanding of the relationship between series and integrals. This understanding is crucial for tackling a wide range of problems involving series convergence, which forms a foundation for further studies in mathematical analysis.

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