Show the next number is not rational

Question

I dont have any idea to prove this number is not rational

$\frac{1}{7}+\frac{1}{7 ^{2}}+\frac{1}{7 ^{6}}+...+\frac{1}{7 ^{n!}}+...$

Please help.

Answer 1

$$\frac{1}{7}+\frac{1}{7(2)}+\frac{1}{7(6)}+...+\frac{1}{7(n!)}+...$$

$$=\sum_{k=1}^{\infty}\frac{1}{7(k!)}$$

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