Show $\sum_{l=0}^\infty \sum_{i=0}^\infty x^{{2^l}(2i+1)}=\sum_{j=1}^\infty x^j$ 
Show $\sum_{l=0}^\infty \sum_{i=0}^\infty x^{{2^l}(2i+1)}=\sum_{j=1}^\infty x^j$ where $|x|<1$

Thus show it is the geometric series.
I know, I could just write out the sum and it would make sense, but is there a more formal proof for this equality?

Rewrite your LHS as
$$
\sum_{\alpha\in A}x^{\alpha}
$$
where
$$
A:=\{2^l(2i+1)\;:\;l,i\in\Bbb N_0\}
$$
and observe that
$$
A=\Bbb N
$$
where $\Bbb N$ and $\Bbb N_0$ are natural numbers starting from $1$ and $0$ respectively.
