Power Series Representation

**mathwiz2006** · November 9th 2008, 02:07 PM

I have to find a power series representation for the following:

1) $1/(1-x^3)$
2) $3x^2/(1-x^3)^2$
3) $-3x^2/(1-x^3)$
4) $ln(1-x^3)$

Well, I know that $1/(1-x) = 1 + x + x^2 + x^3 + x^4 +... =$ the sum of $x^n$
Therefore $1/(1-x^3) = 1 + x^3 + x^6 + x^9 +...=$ the sum of x^(3n)

But I don't know how to relate the others to each other they look similar, so any hins would be greatly appreciated. Thanks!

**Krizalid** · November 9th 2008, 02:12 PM

$\frac{1}{1-x^{3}}=\sum\limits_{k=0}^{\infty }{x^{3k}}.$ So, why not differentiating this result and see what you get?

As for 4), you're expected to know that $\ln (1-x)=-\sum\limits_{k=1}^{\infty }{\frac{x^{k}}{k}},$ then what would be a series representation for $\ln(1-x^3)$ ?

**skeeter** · November 9th 2008, 02:13 PM

note that #2 is the derivative of #1 ...

#4 is the antiderivative of #3

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