MathDB

Prove that numbers of the form

\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,

where

0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)

are rational if and only if starting from some

i

on all the

a_{i}

's are either equal to

0

( in which case the sum is finite) or all are equal to

i-1

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