If $a_i=0$ for all large enough $i$, then $e$, being transcendental, cannot be a solution to your equation, unless $a_i=0$ for all $i\ge1$. Otherwise, if $a_i\ne0$ for infinitely many $i$, then the series for $e$ cannot converge, given that the $a_i$'s are integers, because then $|a_ie^i|\ge e^i\not\to0$ if $a_i\ne0$.
Iosif Pinelis
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