I was trying to solve $e^x = x$ and after expanding: $1+x^2/2!+x^3/3!+x^4/4! ... = 0$ I don't know what to do. Right now I am interested in 2 things: Is there a way to find a closed form for the 0's for the infinite expression, maybe even factor it? (Clearly not in the reals) and second: Within the reals is there a general solution for $e^{a+x} = bx$. I am not looking for any approximations because anyone can do that.Thanks in advance.
2026-04-12 17:51:41.1776016301
Find the set of solutions to a infinite polynomial
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I guess you are looking for the real solutions. By the expansion, for $x\geq 0$, $$e^x-x=1+x^2/2!+x^3/3!+x^4/4!+\dots\geq 1$$ Moreover for $x<0$, $0<e^x=x<0$, so there are no real solutions.
P.S. Over $\mathbb{C}$, the solutions of $\exp(z)=z$ are infinite and they are related with the Lambert W functions (no closed form unfortunately). See the green dots in this picture taken from an interesting paper by Stanislav Sykora.