Can we get series form solutions for polynomial equations of $\text{deg}≥5$

Question

The Abel-Ruffini Theorem states that there is no algebric solution (solution in radicals) to the general polynomial equations of $\text{deg}≥5$. My question begins here: can we have a way to get series form solutions of such equations? I think it is possible for if the numerical methods (like Newton's method) are efficient to unlimited accuracy , then we may combine the numerical methods with coefficient relations of the polynomial to get an infinite series form solution. My question may rather look as just a speculation, but I have asked this question after much thinking, what I need is some rigorous theoretical treatment of the problem. Any help would be appreciated. (This is much like a series representation of some root finding algorithm.)

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I am not asking about the already known methods , rather I want to know if there are solutions of form $\sum^∞_1 a_{i,k}$ (not closed form) where $k=1,2,...,n$for general equations $P(x)=0$ where $\text{deg} (P)≥5$. This is quite a different thing.

Possible duplicate of How to solve an nth degree polynomial equation — Winther, Dec 12 '18 at 10:20
@Winther, this is not a duplicate, I have edited the question, please consider it again. — Awe Kumar Jha, Dec 12 '18 at 10:34
I thought you were interested in learning how to solve such equations and that link has alot of info. The answer to your idea: "then we may combine the numerical methods with coefficient relations of the polynomial to get an infinite series form solution" is no. An $n$th degree equation has $n$ roots. An series can only converge to one solution. When using Newton's method we need to apply it with different "seeds" to find all solutions. Thus a "nice looking" series in the coefficients finding all roots is something I doubt you'll find. — Winther, Dec 12 '18 at 10:37
There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions? — Winther, Dec 12 '18 at 10:40
@Winther, I don't agree with you, terms occupied with the kth roots of unity can provide k different solutions for different values of k in the same series. — Awe Kumar Jha, Dec 12 '18 at 10:45

score 1 · Accepted Answer · answered Dec 12 '18 at 10:37

It is possible, but it is quite boring, I'm afraid. Suppose that the sequence $x_n$ converges to a solution of an equation. Then, letting $x_0:=0$, the series $$ \sum_{n=1}^\infty (x_n-x_{n-1}) $$ trivially converges to the same solution, thus giving a "series form" for it.

In particular, any iterative method, that produces a solution in form of limit of some sequence, can be written in series form.

Can we get series form solutions for polynomial equations of $\text{deg}≥5$

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