Evaluate the improper integral if it converges, or state that it diverges.
E) diverges

Question

Evaluate the improper integral if it converges, or state that it diverges. ​  ​
E) diverges

Quizplus · Accepted Answer

We can use the Limit Comparison Test with the function f(x) = 1/x^2. 
Since the integrand g(x) = 1/(x-1)^2 < 1/x^2 for x > 1, we have 
∫(1 to ∞) 1/(x-1)^2 dx ≤ ∫(1 to ∞) 1/x^2 dx. 
The integral of 1/x^2 converges (p-test with p=2), so by the Comparison Test, the integral of 1/(x-1)^2 also converges. Thus, the answer is D.

Evaluate the Improper Integral If It Converges, or State That