Convergence of Improper Integral

Given the improper integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{1}{x^p} \, dx$, determine if it is convergent or divergent for different values of $p$.

Improper integrals are a fascinating extension of the concept of a definite integral, dealing with integrals where either the interval is infinite or the integrand behaves indefinitely. In this particular problem, the integral spans from a finite number to infinity. Such integrals challenge us to extend the notion of computing "area" to infinite stretches along the function. This demands a conceptual expansion of the properties of integrals that we use for bounded functions over a finite interval.

To address the convergence of the integral with respect to different values of $p$, one employs the principle of comparing the magnitude of the integrand to known benchmark integrals that decidedly converge or diverge. Often, the integral of $1/x$ is a pivotal comparison since it diverges, serving as a critical threshold in many assessment cases. Understanding where $p$ situates the integrand in relation to this threshold is crucial.

This problem integrates valuable techniques, such as comparing decay rates of functions, which is a central theme in calculus when tackling improper integrals. These concepts lay the groundwork for deeper explorations into series convergence and the nature of infinite processes, aiding in the development of a robust, intuitive sense of how functions behave under extreme limits.

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