How to prove this equality  $ t(1-t)^{-1}=\sum_{k\geq0} 2^k t^{2^k}(1+t^{2^k})^{-1}$? 
Prove the equality $\quad t(1-t)^{-1}=\sum_{k\geq0} 2^k t^{2^k}(1+t^{2^k})^{-1}$.

I have just tried to use the Taylor's expansion of the left to prove it.But I failed.
I don't know how the $k$ and $2^k$ in the right occur. And this homework appears after some place with the $Jacobi$ $Identity$ in the book $Advanced$ $Combinatorics$(Page 118,EX10 (2)).
Any hints about the proof ?Thank you in advance.

You can use:
$$\frac{t}{t-1} = \sum_{n=1}^\infty{t^n}=t+t^2+(t^3+t^4)+(t^5 + ... + t^8)$$
And then use:
$$\sum_{n=2^k}^{n=2^{k+1}}{t^{n}} = \frac{t^{2^k}(t^{2^{k+1}}-1)}{t-1}$$
