Power Series Representation...

Question

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence

$$ f(x)= x^2\ln(1+x^2) $$

How should I start it out?

Answer 1

Hint

Take the power series of $\log(1+y)$; replace $y$ by $x^2$; multiply the result by $x^2$.

I am sure that you can take from here.

Answer 2

$$1-x^2+x^4-x^6+\cdots=\frac{1}{1+x^2}\tag{by the geometric series}$$

$$2x-2x^3+2x^5-\cdots=\frac{2x}{1+x^2}$$

$$x^2-\frac{x^4}{2}+\frac{x^6}{3}-\cdots=\ln(1+x^2)\tag{integrating from $0$ to $x$}$$

$$x^4-\frac{x^6}{2}+\frac{x^8}{3}-\cdots=x^2\ln(1+x^2)$$

Determining radius of convergence is pretty straightforward here.