# 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?

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Find the power series for $\ln(1+x^2)$ first. You might find the power series for $1/(1-x)$ helpful. – Nameless Mar 21 '14 at 8:56
This is what confuses me... How do I change that into a form of $$1/(1-x)$$ The derivative is $$2x/(1+x^2)$$ – StrugglingCalcStudent Mar 21 '14 at 8:57

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.

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I think I get it... Let me try it out... Thanks – StrugglingCalcStudent Mar 21 '14 at 9:05
@StrugglingCalcStudent. You are welcome and you will get it ! – Claude Leibovici Mar 21 '14 at 9:05

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

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

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

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

Determining radius of convergence is pretty straightforward here.

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