Evaluate this infinite sum $\displaystyle \sum_{n=1}^{\infty} \left(x^{3n-2} - x^{3n+1}\right)$
We can't simplify anything any more. Except.
$\displaystyle \sum_{n=1}^{\infty} \left( \frac{x^{3n}}{x^2} - \frac{x^{3n}x^3}{x^2} \right)$
$\displaystyle \sum_{n=1}^{\infty} \frac{x^{3n}(1 - x^3)}{x^2}$

$\displaystyle S=\sum_{n=1}^{\infty} x^{3n-2} - x^{3n+1}=\dfrac{1-x^3}{x^2}\sum_{n=1}^\infty(x^3)^n$
If $|x^3|\le1,$ using Infinite Geometric Series Formula Derivation,
$\displaystyle S=\frac{1-x^3}{x^2}\cdot\dfrac{x^3}{1-x^3}=x$
