<s>[INST] Power Series Representation... 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?
 [/INST] $$1-x^2+x^4-x^6+\cdots=\frac{1}{1+x^2}\tag{by the geometric series}$$
$$2x-2x^3+2x^5-\cdots=\frac{2x}{1+x^2}$$
$$x^2-\frac{x^4}{2}+\frac{x^6}{3}-\cdots=\ln(1+x^2)\tag{integrating from $0$ to $x$}$$
$$x^4-\frac{x^6}{2}+\frac{x^8}{3}-\cdots=x^2\ln(1+x^2)$$
Determining radius of convergence is pretty straightforward here.
</s>