Solution

Find the first several terms of the sequence associated with the simple continued fraction representation for $e$.


While it doesn't repeat, there is a clear pattern to the simple continued fraction for $e$:

$$e = [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,\ldots]$$

Interestingly, proving this pattern continues to hold also establishes the irrationality of $e$, as all rational values have a finite simple continued fraction.