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