Why is the correction important?

[Soccerdude]

Homework Statement

Find the Taylor series of f(x) = x²ln(1+2x²) centered at c = 0.

Homework Equations

Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)^n-1xⁿ/n

The Attempt at a Solution

I have worked the problem to

(-1)ⁿ4ⁿx²ⁿ/n

I am not sure where to go from here

Thank You!

 

[SammyS]

Homework Statement

Find the Taylor series of f(x) = x²ln(1+2x²) centered at c = 0.

Homework Equations

Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)^n-1xⁿ/n

The Attempt at a Solution

I have worked the problem to

(-1)ⁿ4ⁿx²ⁿ/n

I am not sure where to go from here

Thank You!

Well, you seem to have the right idea for the Taylor series of ln(1+2x²). It's not quite right.

$\displaystyle \left(2x^2\right)^n=2^nx^{2n}\ .$

When you make the correction for that, then just multiply by x² , to get the Taylor series for x²ln(1+2x²).