Barrons AP Calculus - David Bock

The set of all values of x for which a power series converges is called its interval of
convergence. To find the interval of convergence, first determine the radius of convergence by
applying the Ratio Test to the series of absolute values. Then check each endpoint to determine
whether the series converges or diverges there.

EXAMPLE 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:

SOLUTION:

which is less than 1 if |x − 2| < 2, that is, if 0 < x < 4. Series (4) converges on this interval and