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Example 8: Integral Test

Use the Integral Test to test the improper integral 09_infinite_series-230.gif for convergence.

By Example 7 the series 09_infinite_series-231.gif 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.

Last Update: 2006-11-15