Is there a sequence of rational numbers $a_n$ such that $a_1 e^{-1} + a_2 e^{-2} + cdots = 1$?

Given an arbitrary real number $x in (0,1)$, we can construct a sequence of rational numbers $(a_n)_{n = 1}^{infty}$ such that $displaystylesum_{n = 1}^{infty}a_nx^n = 1$ as follows:

Let $a_1 = leftlfloordfrac{1}{x}rightrfloor$. For each integer $n ge 2$, let $a_n = dfrac{1}{n}leftlfloor dfrac{n}{x^n}left(1-displaystylesum_{k = 1}^{n-1}a_kx^kright) rightrfloor$.

Note that once $a_1,ldots,a_{n-1}$ are chosen, $a_n$ is the largest rational number of the form $dfrac{m}{n}$ such that $displaystylesum_{k = 1}^{n}a_nx^n$ does not exceed $1$. With this construction, $left|1-displaystylesum_{k = 1}^{n}a_kx^kright| le dfrac{1}{n}$ for all integers $n ge 1$, and thus, $displaystylesum_{k = 1}^{infty}a_kx^k = 1$.