Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express $W(x)$ for $\left\{x\in\mathbb{R}:x\geq-\frac{1}{e}\right\}$? I am aware of Newton's method's approximation, $W(x)\approx\lim_{n\to\infty}w_n,w_{n+1}=\frac{xe^{-w_n}+(w_n)^2}{w_n+1}$, but this is not an exact value or a closed form, like the Taylor series. Also, I came across these two questions, Are there other power series for the Lambert W function than this one? and Question about Lambert W function, with comments and answers that seem to answer my question, but I am lost on how they derived them, if they are actually valid, and what the solution's closed forms are. I apologize if I sound unreasonably ignorant in this question. Thanks in advance.

• Series expansion is not generally regarded as "closed form". Strictly speaking, provided a proof of convergence, a Newton iteration or similar is as much of a "closed form" as a series expansion. Both provide a sequence which converges to the desired value. – Ian Jan 10 '15 at 21:40
• @Ian, Sorry, I mixed up the mathematical meaning of closed form when I said it, as you mentioned. As for Newton's method, I learned that it was just an approximation for finding roots and does not actually give you an exact converging value like the Taylor series does. So, I am kind of confused as having a proof of convergence for it. – Emu Jan 10 '15 at 22:02
• Proofs of convergence for Newton's method and other numerical methods are often specific to the problem in question, but we can still prove they work. For example, I can prove to you that the sequence $x_1=1$, $x_{k+1}=\frac{x_k}{2}+\frac{y}{2x_k}$ converges as $k \to \infty$ to $\sqrt{y}$. – Ian Jan 10 '15 at 22:11
• @Ian, Oh, I see. You have been very helpful, and I wouldn't want to bother you anymore, so can you redirect me to a website that can help me understand how to prove convergence for your example and the example in my original question. Thank you. – Emu Jan 10 '15 at 22:32
• @Emu: See this post for a rigorous analysis of the Newton-Raphson approximation method. You can find more example applications of asymptotic analysis linked from my profile. – user21820 Aug 13 '18 at 5:36