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I have been going at this question for weeks now and couldn't find anything. Can we have a series of the form: $$f(x)=\sum_{n=0}^{\infty} a_n x^n$$ where $a_n$ are rationals and not all $0$ such that $f(e)=0$? Just like how $\pi$ is a root of the power series of $\sin(x)$ function: $$\sin(x) = \s...