Question

Find the coefficient of x 4 in the taylor series expansion centered at the origin for the function f(x) = 8 ln(9 + 3x 2 ).

Answer 1

The coefficient of x4 in the Taylor series expansion of the given function is 0.

Step-by-step explanation:

To find the coefficient of x4 in the Taylor series expansion of the function f(x) = 8 ln(9 + 3x2), we need to compute the fourth derivative of f(x) and evaluate it at x = 0.

The Taylor series expansion for a function centered at the origin is given by:

f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + f''''(0)x4/4! + ...

In this case, we have:

f'''(x) = 48x/(9 + 3x2)

Evaluating f'''(x) at x = 0 gives:

f'''(0) = 0/9 = 0

Therefore, the coefficient of x4 in the Taylor series expansion centered at the origin for the function f(x) = 8 ln(9 + 3x2) is 0.

Answer 2

Try this option.
Note, that it is the full solution, it is possible to make it shorter.