Final answer:
The convergence of the integral ∫(a to [infinity]) f(x)dx doesn't provide enough information to conclude the convergence of another integral ∫(a to [infinity]) g(x)dx, unless there is a specific relationship between the two functions. Thus, the correct answer is that it depends on the functions.
Step-by-step explanation:
The question asks whether the convergence of the integral ∫(a to [infinity]) f(x)dx implies anything about the convergence of the integral ∫(a to [infinity]) g(x)dx. The convergence of an integral from a to infinity refers to whether the area under the curve of the function from point a to infinity has a finite value.
In this scenario, knowing that ∫(a to [infinity]) f(x)dx is convergent doesn't provide enough information to determine the convergence of ∫(a to [infinity]) g(x)dx. The convergence of one integral does not imply the convergence of another unless there is some known relationship between f(x) and g(x), such as g(x) being bounded by a multiple of f(x) or both functions having similar properties that affect their respective integrals’ convergence.
Therefore, the correct answer to the question is: D) Depends on the functions. The convergence of one integral does not guarantee the convergence of another unrelated integral.