Barrons AP Calculus - David Bock

diverges if |x − 2| > 2, that is, if x < 0 or x > 4.

When x = 0, (4) is 1 − 1 + 1 − 1 + · · · and diverges. When x = 4, (4) is 1 + 1 + 1 + · · · and

diverges. Thus, the interval of convergence is 0 < x < 4.

EXAMPLE 37

Find all x for which the series converges.

SOLUTION: converges only at x = 0, since

unless x = 0.

C2. Functions Defined by Power Series.

Let the function f be defined by

its domain is the interval of convergence of the series.
Functions defined by power series behave very much like polynomials, as indicated by the
following properties:

PROPERTY 2a. The function defined by (1) is continuous for each x in the interval of convergence
of the series.
PROPERTY 2b. The series formed by differentiating the terms of series (1) converges to f ′(x) for
each x within the radius of convergence of (1); that is,

Note that power series (1) and its derived series (2) have the same radius of convergence but not
necessarily the same interval of convergence.

EXAMPLE 38

Let

Find the intervals of convergence of the power series for f (x) and f ′(x).

SOLUTION:

also,