Formal proof that $e^x$ is not algebraic

Question

How do I give a formal proof that $e^x$ is not algebraic, like for example: $$\sum_{n\geq0}\frac{x^n}{n!}\notin\mathbb{C}_{\mathrm{alg}}[[x]]$$ Help appreciated!

Answer 1

The notation $\mathbb C_{\mathrm{alg}}[[x]]$ is a bit unorthodox, but I presume what is meant is the algebraic closure of $\mathbb C(x)$ in $\mathbb C((x)).$

One way is to think about the rate of growth at infinity. Suppose that $e^{nx} + p_{n-1}(x) e^{(n-1)x} + \cdots + p_1(x) e^x + p_0(x) = 0$ for some rational functions $p_i$. Now divide by $e^{nx}$, to get $1 + p_{n-1}(x) e^{-x} + \cdots + p_1(x)e^{-(n-1) x} + p_0(x) e^{-n x} = 0.$ Taking the limit as $x \to \infty$, using the fact that exponential decay beats the growth of any rational function, we get that $ 1 = 0$, a contradiction.

(This is related to the fact that $e^x$, thought of as a complex function, has an essential singularity at infinity, unlike algebraic functions.)