converges.Example 14 __
Does converge?
SOLUTION: The associated improper integral is
which equals
The improper integral and the infinite series both diverge.
Example 15 __
Test the series for convergence.
SOLUTION:
by an application of L’Hôpital’s Rule. Thus converges.
THE p-SERIES TEST
A p-series converges if p > 1, but diverges if p ≤ 1.
This follows immediately from the Integral Test and the behavior of improper
integrals of the form .
Example 16 __
Does the series . . . converge or diverge?
SOLUTION: The series . . . is a p-series with p = 3; hence the
series converges by the p-Series Test.