Barrons AP Calculus - David Bock

earlier, this is a necessary condition for convergence, but may not be sufficient. We now examine the
sum using the same technique we employed in Example 10:

EXAMPLE 13

Does 0.3 + 0.03 + 0.003 + · · · converge or diverge?

SOLUTION: The series 0.3 + 0.03 + 0.003 + · · · is geometric with a = 0.3 and r = 0.1. Since |r| <

1, the series converges, and its sum is

NOTE: = 0.333 ..., which is the given series.

B4. Tests for Convergence of Nonnegative Series.

The series is called a nonnegative series if an ≥ 0 for all n.

THE INTEGRAL TEST
If f (x) is a continuous, positive, decreasing function and f (n) = an, then converges if and only if
the improper integral 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.