Improper Integral Convergence using Comparison Test

Determine if the integral $\displaystyle \int_{1}^{\infty} \frac{1}{1+x} \, dx$ is convergent or divergent using the comparison test.

In this problem, we explore the behavior of an improper integral, particularly focusing on whether it converges or diverges. The comparison test is a powerful tool in this analysis. By comparing the given function with a second function that has a known convergence behavior, we determine the properties of the original integral. This method requires us to choose a comparator function that is easier to evaluate and has similar characteristics to the function in question as the input approaches infinity. This approach to analysis hinges on our ability to understand limits and comparisons, honing our skills in selecting appropriate functions for practical evaluation.

The integral in question is defined across an infinite domain, a typical scenario in improper integrals. The concept of convergence needs us to evaluate whether this integral settles to a finite value or spirals into an infinite one. Understanding the convergence of integrals over infinite intervals is crucial in advanced calculus and real analysis, as these concepts frequently appear in mathematical modeling and theoretical developments. Mastery of such tests not only strengthens foundational calculus skills but also prepares students for more complex applications in analysis, physics, and engineering.

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