The sereis $ \sum_{n=1}^{\infty} 1/2^{n^2} $ converges to irrational? 
Does the sereis $\sum_{n=1}^{\infty} 1/2^{n^2} $ converges to an irrational ?

Thinking about $\sum_{n=1}^{\infty} 1/10^{n^2} $ , we can say it is an irrational number due to non-recurring decimal places.
Does this argument work for the above sereis considering binary representations? Is it that simple or is there a more analytical way? Please help. 

HINT:
If $\alpha = \frac{p}{q}$ is a rational number, then for any $k$ natural number,  $k \cdot \alpha$ cannot get closer than $\frac{1}{q}$ to an integer. 
Now we have 
$$2^{n^2} \alpha = \textrm{integer} + 2^{n^2}\sum_{k=n+1}^{\infty} \frac{1}{2^{k^2}}= \textrm{integer} + \delta_n $$
where $0<\delta_n \to 0$ as $n\to \infty$.
