Does there exist a positive real number $x$, whose decimal expansion is non-terminating, such that:
$\large \displaystyle x = \overline{a_0.a_1a_2a_3a_4a_5\cdots} = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cfrac{1}{a_5 \, + _\ddots}} } } }$
$\large \displaystyle a_k$ maps each $\large k$ to a decimal digit.
I have not been able to find any references or similar topics out there on continued fractions.
My question also extends to arbitrary integer bases and possibly non-integer bases.