Finding a3 in the Taylor Series for x^3ln(1+x^2)

[lmannoia]

Homework Statement

Let f(x) = x³ln(1+x²), and let the summation (from n=0 to infinity) a_nxⁿ be the Taylor series for f about 0. Then what is a₃?

Homework Equations

The Attempt at a Solution

What?! I definitely don't expect the answer, but does anyone know how I could go about finding this out from the vague information given about the summation?

 

[Mark44]

Homework Statement

Let f(x) = x³ln(1+x²), and let the summation (from n=0 to infinity) a_nxⁿ be the Taylor series for f about 0. Then what is a₃?

Homework Equations

The Attempt at a Solution

What?! I definitely don't expect the answer, but does anyone know how I could go about finding this out from the vague information given about the summation?

How do you normally go about finding the coefficients of the terms in a Taylor's series? In this case, it's a Maclaurin series.

 

[lmannoia]

It would be f(0)+f1(0)x+f2(0)x^2/2! +f3(0)x^3/3!.. what I'm not getting is, how can I do this when I'd be plugging in zero? Wouldn't all of the coefficients just be zero? Or am I thinking about this in the wrong way?

 

[Mark44]

What do you mean by f1(0)? f2(0)? f3(0)? Do you mean f'(0), f''(0), and f'''(0)?

Clearly f(0) = 0, and it turns out that f'(0) = 0. Have you determined what f''(x) and f'''(x) are?