Generalized case for identifying transcendental numbers

Question

For any strictly increasing sequence of natural numbers $ a_1...a_n $, is the expression given below transcendental? Stack Exchange gives a specific case Is $0.1010010001000010000010000001 \ldots$ transcendental? , with $ a_n=n$, as transcendent, but people I've asked don't think that can be generalized to all cases, likewise with using Liouville's inequality.

$ \sum_{n=1}^{\infty} \prod_{i=1}^{n} 10^{-a_i} $

Answer 1

Modern mathematics knows very little about transcendental numbers. The closest thing we have to a general case is $a^b$ where $a$ is algebraic but not $0$ or $1$, and $b$ is irrational algebraic (by the Gelfond-Schneider theorem), in particular: $2^{\sqrt{2}}$, the Gelfond-Schneider constant (or Hilbert number).