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

We can use the integral test to determine whether the improper integral converges or diverges. Let f(x) = xe^(-x). Then f(x) is continuous, positive, and decreasing for x ≥ 1. Therefore,

∫(1 to ∞) f(x) dx converges if and only if the series ∑(n=1 to ∞) f(n) converges.

Now,

∑(n=1 to ∞) f(n) = ∑(n=1 to ∞) n e^(-n)

By the ratio test,

lim(n→∞) |(n+1) e^(-(n+1))/n e^(-n)| = lim(n→∞) (1/n) = 0

Therefore, the series ∑(n=1 to ∞) n e^(-n) converges. So, the integral ∫(1 to ∞) xe^(-x) dx also converges.

Thus, the answer is A, the integral converges.

Determine Whether the Improper Integral Diverges or Converges