# Prove an infinite sum is irrational [duplicate]

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.

## marked as duplicate by Hans Lundmark, Dando18, Daniel W. Farlow, Siong Thye Goh, Claude LeiboviciAug 3 '17 at 6:45

$$\sum_{k=1}^{\infty} 7^{-k!} = \frac{1}{7} + \frac{1}{7^{2!}} + \frac{1}{7^{3!}} + \dots$$ 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.