Periodic representation of pi in varying base system

Question

Pi and e ar irrational numbers and cannot have a periodic representation in a fixed base number (binary, decimal, hex, etc). However, if you choose variable base like 1!, 2!, 3!,.. e becomes 1.1111..1.. there is a similar trick to select a “variable base” to make pi=3.222..2... (or was it 2.2222...2... ?) but I don’t remember the definition of the base. What works of be the rule for the base and how to find such rules for arbitrary numbers (say Euler constant 0.577...)?

Answer 1

According to this logic:

Since $e=1+\frac{1}{2!}+\frac{1}{3!}+\cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111\ldots$.

Then $\pi=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots$. So choose "base $4$, $\frac 43$, $\frac 45$, etc., to write $\pi$ as

$$1.010101010\cdots-(0.101010\ldots)$$