Prove an infinite sum is irrational

Question

I'm trying to prove that

$$ \sum_{k=1}^{\infty} 7^{-k!} $$

is irrational but I'm so lost. Any tips for where to begin, thanks in advance.

Answer 1

$$\sum_{k=1}^{\infty} 7^{-k!} = \frac{1}{7} + \frac{1}{7^{2!}} + \frac{1}{7^{3!}} + \dots$$ has a base 7 representation of $(0.11000100.....1000000.............1000000000......)_7$ where there is a $1$ at every $n!$th place from the radix point, and $0$s at the rest of the places.

A real number is rational if and only if its positional representation either terminates or repeats in any base.

This series converges to a number whose base 7 representation does not repeat (clearly), or terminate. Therefore, it is irrational.