Let $r \in \Bbb R$ be such that $0 < r < 1$. Prove that the series
$r^2 + r + r^4 + r^3 + ... + r^{2n} + r^{2n-1} + ...$
converges or diverges.
I know $\sum_{n=0}^{\infty} r^n$ converges to $\frac{1}{1-r}$ for $|r| <1$. I'd like to rearrange the series above to be written as
$r + r^2 + r^3 + r^4 +...+ r^{2n-1} + r^{2n} + ...$
Is that all I would need to prove the series in question converges? Compare it to $\sum_{n=0}^{\infty} r^n$ beginning at $n = 1$ instead of $n = 0$ ?