Barrons AP Calculus - David Bock

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.

EXAMPLE 17

Does the series converge or diverge?

SOLUTION: diverges, because it is a p-series with

THE COMPARISON TEST

We compare the general term of the nonnegative series we are investigating, with the general
term of a series known to converge or diverge.
(1) If converges and an un, then converges.
(2) If diverges and an un, then diverges.
Any known series can be used for comparison. Particularly useful are p-series, which converge if
p > 1 but diverge if p 1, and geometric series, which converge if |r| < 1 but diverge if |r| 1.

EXAMPLE 18

Does converge or diverge?

SOLUTION: Since and the p-series converges, converges by the

Comparison Test.

EXAMPLE 19

Does the series converge or diverge?