Barrons AP Calculus - David Bock

and

Hence, the power series for f converges if −1 x 1.

For the derivative

also,

and

Hence, the power series for f ′ converges if −1 x < 1.
Thus, the series given for f (x) and f ′(x) have the same radius of convergence, but their intervals
of convergence differ.

PROPERTY 2c. The series obtained by integrating the terms of the given series (1) converges to

for each x within the interval of convergence of (1); that is,

EXAMPLE 39

Let Show that the power series for converges for all values of x 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