Determine that a series is rational

Question

Determine whether $$\sum_{n=1}^\infty 1/10^{n!} $$ is rational. I have tried thinking about decimal representations such as that of $1/11$, and the fact that this sum is equal to $0.1100010....1...........1$ etc, but I don't know if the distance between $1$'s increases fast enough (or if it even matters) for this to converge to a rational number.

Answer 1

That sequence of digits is not quasi-periodic. Therefore, that number is irrational. There are many similar examples, such as $\displaystyle\sum_{n=1}^\infty\frac1{10^{n^2}}$.