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?
The question was asked, and a reference to a proof of transcendentality was given, at the accepted answer here at MO.
Actually, it seems it has also been answered here at m.se.