12 __
Does converge or diverge?
SOLUTION: Since , the series diverges by the nth Term
Test.
THE GEOMETRIC SERIES TEST
A geometric series converges if and only if |r| < 1.
If |r| < 1, the sum of the series is .
The series cannot converge unless it passes the nth Term Test; only if
|r| < 1. As noted 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: . . . , which is the given series.
BC ONLYB4. Tests for Convergence of Nonnegative Series
The series is called a nonnegative series if an ≥ 0 for all n.
THE INTEGRAL TEST
Let be a nonnegative series. If f(x) is a continuous, positive, decreasing
function and f(n) = an, then converges if and only if the improper integral