Determine whether the improper integral diverges or converges.Evaluate the integral if it converges.
E)diverges

Question

Determine whether the improper integral  diverges or converges.Evaluate the integral if it converges. ​
E)diverges

Quizplus · Accepted Answer

This integral can be solved using the limit comparison test with the function f(x) = 1/x. Specifically, we have:

lim (x→∞) (e^(1/x)) / (1/x) = 1

Since this limit is finite and positive, we can apply the limit comparison test and conclude that the given integral converges if and only if the integral of 1/x converges. However, we know that the integral of 1/x over the range [1, ∞) diverges (e.g. by the p-test), so the given integral must also diverge.

Of the answer choices, only (B) matches this conclusion.

Determine Whether the Improper Integral Diverges or Converges