Convergence or Divergence of Series Using Integral Test

Determine whether the series represented by the function is convergent or divergent using the integral test for the function: $\displaystyle \int_{1}^{\infty} \frac{x}{x^2 + 1} \, dx$

When tasked with determining the convergence or divergence of a series such as this, it's useful to employ the integral test, which bridges the concepts of improper integrals and infinite series. The test essentially provides a criterion where an improper integral that converges implies the convergence of the corresponding series and vice versa. This approach standardizes the process for addressing a specific class of numerical series and is especially useful for series that can be expressed in terms of a positive, continuous, decreasing function.

Exploring this particular problem, the function involved requires us to evaluate an improper integral. Improper integrals are those where the interval of integration is infinite or the integrand approaches infinity within the interval. In solving, attention will be focused on the limits of integration leading toward infinity. Establishing convergence involves solving the integral from a lower limit to indefinite infinity, analyzing its integral behavior.

Understanding this problem offers insight into the use of calculus to solve series problems—an intersection of analysis tools. It requires a synthesis of recognizing when to apply certain tests and the mathematical skills from multiple techniques in integration. This knowledge is foundational for those progressing into advanced calculus and analysis courses.

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