# Sum Things Are Easier Than They Appear

Algebra Level 3

$\large \alpha = \sum_{i=1}^\infty \frac 1{2^i} + \sum_{j=1}^\infty \frac 1{2^{j+1}} + \sum_{k=1}^\infty \frac 1{2^{k+2}} + \cdots$

Let $$\alpha$$ as defined above. Find $$\displaystyle \sum_{n=1}^\infty \frac 1{2^{n \alpha}}$$.