Evaluate the Integral and Determine Convergence

Evaluate the integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{1}{x} \, dx$ and determine if it is convergent or divergent.

When evaluating an integral from a specific point to infinity, it's categorized as an improper integral, often because the region over which we're integrating is unbounded. For this particular problem, the given integral involves the function one over x, which is a crucial example to understand when dealing with improper integrals. The primary strategy involves determining the behavior of the integral as x approaches infinity, and this is done by calculating the limit of the integral as its upper bound tends toward infinity.

In broader terms, an integral like this demonstrates the importance of evaluating whether an integral converges to a finite value or diverges to infinity. This question inherently addresses the distinction between convergent and divergent behaviors, which is fundamental when considering the integrals of unbounded regions. Within calculus, this involves not only calculation but understanding the conceptual underpinnings that govern such behaviors.

Related Problems