Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational?
Clearly this series is convergent (compare to geometric series with ratio
1/2). I'm sure it's irrational since a rational number written in base 2
will have either a terminating or repeating decimal representation. But
the hard part is to show this representation in question doesn't repeat.
(cf
https://www.google.com/search?q=periodic+rational+base&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a
)
Can you show this number is transcendental?