Question

Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that has a power series expan sion. Do not show that
Rn(x) -----> 0 .] Also find the associated radius of convergence.

Answer 1

Step 1:

The function is .

Maclaurin series is

Find consecutive derivatives of the function to know the pattern of the n th derivative of the function.

Differentiate with respect to on each side.

Similarly, we can write .

Find the values of the above functions at 0.

.

Step 2:

Substitute above values in the Maclaurin series formula.

Maclaurin series of the function is

Find the radius of convergence using ratio test.

The series is  .

Consider and

By the ratio test the series is convergent when .

Hence the radius of convergence is .

Solution:

Maclaurin series of the function is .

Radius of convergence is .