Champernowne’s constant

For a given base $b$, Champernowne’s constant $C_b$ is the result of concatenating the base $b$ digits of the positive integers in order after 0 and a decimal point, that is,

$$\sum _{i=1}^{\mathrm{\infty }}\frac{i}{b^{{\sum }_{j=1}^{i}k}}$$

(where $k$ is the number of digits of $j$ in base $b$).

Kurt Mahler proved that $C_{10}$ (approximately 0.123456789101112131415161718192021…) is a transcendental number. Champernowne had earlier proved that $C_{10}$ is a normal number.