Convergence of Integral from 1 to Infinity Using Comparison Theorem

Determine the convergence or divergence of the integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{2 + e^{-x}}{x} \, dx$ using the comparison theorem.

The comparison theorem is a powerful tool used to determine the convergence or divergence of improper integrals by comparing the integral in question to another integral whose behavior is well understood. This technique relies on establishing an inequality between the function being integrated and another function that bounds it from above or below on a given interval, particularly as it approaches the point of discontinuity or infinity. In the context of integrals, the comparison theorem is particularly useful when dealing with rational or semi-rational expressions that may not be easily integrated or for which integration leads to complicated expressions.

In applying the comparison theorem, the key strategy is to identify a function with known convergence or divergence behavior that closely resembles the function within the integrand in terms of growth rates or dominant terms. For the integral of $\frac{2 + e^{-x}}{x}$ from 1 to infinity, consider breaking the integrand into simpler components. The constant term $\frac{2}{x}$ suggests comparison with the harmonic series or its integral counterpart, which is known to diverge. Meanwhile, the term involving $\frac{e^{-x}}{x}$ can often be compared to integrals of exponential decay functions, which typically converge rapidly.

Analyzing this integral involves understanding the dominance of the constant term over the decaying exponential term as $x$ approaches infinity. The behavior of each component over the interval contributes to determining the overall convergence. Ultimately, because the term $\frac{2}{x}$ diverges, it indicates potential divergence for the entire integral unless a convergent bounding function can be identified through another thoughtful application of comparison methods. Conceptual understanding of why certain terms dominate others in improper integrals helps in making informed decisions about which comparisons are valid and revealing about the problem’s convergence properties.

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