Consider the following power series:
$f(x) = \sum\limits_{i>=1} 2^{i-1}x^{3i}$ = $\ x^3 + 2x^6 + 4x^9 + ...$
$g(x) = \sum\limits_{i=2}^{20} f(x)^{i}$
Express both f(x) and g(x) as rational functions, ie p(x)/q(x) where P(x) and q(x) are polynomials.
How would I go about doing this? Is there some kind of algorithm or pattern I can follow?
Hints:
$$2^{i-1}x^{3i}=\frac12\left(2x^3\right)^i$$
So assuming
$$2|x|^3<1\;\;\text{(why?)}\;,\;\;\text{we get}\;\;\sum_{k=1}^\infty2^{i-1}x^{3i}=\frac12\frac{2x^3}{1-2x^3}$$