BC ONLYExample 33 __
Find all x for which the following series converges:
SOLUTION: By the Ratio Test, the series converges if
Thus, the radius of convergence is 1. The endpoints must be tested separately
since the Ratio Test fails when the limit equals 1. When x = 1, (3) becomes 1 + 1
- 1 + . . . and diverges; when x = −1, (3) becomes 1 − 1 + 1 − 1 + . . . and
diverges. Thus the interval of convergence is −1 < x < 1.
Example 34 __
For what x does converge?
SOLUTION:.
The radius of convergence is 1. When x = 1, we have . . . , an
alternating convergent series; when x = −1, the series is . . . , which
diverges. Thus, the series converges if −1 < x 1.
Example 35 __
For what values of x does converge?
SOLUTION:
which is always less than 1. Thus the series converges for all x.
Example 36 __
Find all x for which the following series converges: