Power series for derivatives

a. Differentiate the Taylor series about 0 for the following functions.

b. Identify the. function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.

f(x) = In (1 + x)

Solution 23E

Step 1:

In this problem we have to find the power series for derivatives.

a. Differentiate the Taylor series about 0 for the following functions.

We have

The taylor series for with center 0 is

Differentiate, we get

Step 2:

b. Identify the function represented by the differentiated series.

The differentiated series is which is

Therefore the function represented by the differentiated series of is .