Example 8: Integral Test

Use the Integral Test to test the improper integral for convergence.

By Example 7 the series converges.

For x > 1 the function

f(x) = (ln x)/x²

is continuous, positive, and has derivative

f'(x) = x^-3(1 - 2 ln x).

Thus for x > √e >, f'(x) < 0 and f(x) is decreasing. Therefore the Integral Test applies and the improper integral converges.