in the interval of convergence of the power series for f(x).
SOLUTION: Obtain a series for by long division.
Then,
It can be shown that the interval of convergence is −1 < x < 1.
Then by Property 2c
BC ONLYSince when x = 0 we see that c = 1, we have
Note that this is a geometric series with ratio r = x and with a = 1; if |x|< 1, its
sum is .
C3. Finding a Power Series for a Function: Taylor and Maclaurin
Series
If a function f(x) is representable by a power series of the form c 0 + c 1 (x − a) +
c 2 (x − a)^2 + . . . + cn(x − a)n + . . .
on an interval |x − a| < r, then the coefficients are given by Taylor Series
The series