TITLE: Is this number rational?
QUESTION [13 upvotes]: Consider the following number:
$x=0.01001000100001...$
(the number of '$0$' between '$1$'s increases linearly)
given by the sum
$$x=\sum_{i=1}^\infty 10^{-i - i (1 + i)/2}$$
My question is whether this number (or more generally, numbers generated by this technique), are rational or irrational, and what is the sketch of proof.

REPLY [12 votes]: Perhaps it could be mentioned that if you space out more drastically your ones
$$
\sum_{i=1}^{\infty} 10^{- i!}
$$
you get Liouville's number which is transcendental over the rationals.