Prove the series $\sum_{n=1}^{\infty}\frac{1}{{5^n}^!}$ converges to an irrational number
Solution 1:
Because in base $5$ that number will have a non-periodic expansion.
You can also prove it directly by using that the distance between non-zero digits increases. This means that you can for every $\epsilon>0$ find an (non-zero) integer $a$ such that $a\sum 5^{-n!}$ can be written as $N+\xi$ where $N$ is a natural number and $0<\xi<\epsilon$. If $k\sum 5^{-n!}$ is a natural number then so is $ak\sum 5^{-n!} = kN + k\xi$ where $0<\xi<k\epsilon<1$. Similar argument can be used to show that the number is transcendental.