Barrons AP Calculus

Example 38 __

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

SOLUTION:

also,

and

BC  ONLY

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 dt 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