Final answer:
To assess if the improper integral ∫ dx/(1-3x) converges or diverges, we would need specific limits of integration to evaluate the antiderivative, which is not provided here. Without the limits, we cannot determine the convergence or divergence of the integral.
Step-by-step explanation:
To determine if the improper integral ∫ dx/(1-3x) converges or diverges, we can evaluate it using an antiderivative. By finding an antiderivative of the integrand, we can then take limits to assess the behavior of the integral. The antiderivative of 1/(1-3x) is -(1/3) × ln|1-3x|. We would typically evaluate this from a to b, as the limits approach infinity or negative infinity depending on the bounds of the integral.
However, in this case, we don't have specific limits, which would be necessary to calculate a definite integral. Improper integrals can have infinite limits or can involve integrands with unlimited growth at certain points in their domain. Without limits of integration, we cannot definitively state whether the integral converges or diverges.
The usual approach to an improper integral would involve evaluating the limit as the bounds approach their respective infinity, but we're missing this information. Therefore, we can only say that to assess convergence, we need to know the specific bounds that the integral is being evaluated over.