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












0












$begingroup$


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.)



Edit



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.






algebra-precalculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    Possible duplicate of How to solve an nth degree polynomial equation
    $endgroup$
    – Winther
    Dec 12 '18 at 10:20










  • $begingroup$
    @Winther, this is not a duplicate, I have edited the question, please consider it again.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:34










  • $begingroup$
    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.
    $endgroup$
    – Winther
    Dec 12 '18 at 10:37












  • $begingroup$
    There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
    $endgroup$
    – Winther
    Dec 12 '18 at 10:40










  • $begingroup$
    @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.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:45


















0












$begingroup$


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.)



Edit



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.






algebra-precalculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    Possible duplicate of How to solve an nth degree polynomial equation
    $endgroup$
    – Winther
    Dec 12 '18 at 10:20










  • $begingroup$
    @Winther, this is not a duplicate, I have edited the question, please consider it again.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:34










  • $begingroup$
    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.
    $endgroup$
    – Winther
    Dec 12 '18 at 10:37












  • $begingroup$
    There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
    $endgroup$
    – Winther
    Dec 12 '18 at 10:40










  • $begingroup$
    @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.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:45
















0












0








0





$begingroup$


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.)



Edit



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.






algebra-precalculus







share|cite|improve this question











$endgroup$




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.)



Edit



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.






algebra-precalculus




algebra-precalculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 10:41







Awe Kumar Jha

















asked Dec 12 '18 at 10:15









Awe Kumar JhaAwe Kumar Jha

40813




40813












  • $begingroup$
    Possible duplicate of How to solve an nth degree polynomial equation
    $endgroup$
    – Winther
    Dec 12 '18 at 10:20










  • $begingroup$
    @Winther, this is not a duplicate, I have edited the question, please consider it again.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:34










  • $begingroup$
    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.
    $endgroup$
    – Winther
    Dec 12 '18 at 10:37












  • $begingroup$
    There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
    $endgroup$
    – Winther
    Dec 12 '18 at 10:40










  • $begingroup$
    @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.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:45




















  • $begingroup$
    Possible duplicate of How to solve an nth degree polynomial equation
    $endgroup$
    – Winther
    Dec 12 '18 at 10:20










  • $begingroup$
    @Winther, this is not a duplicate, I have edited the question, please consider it again.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:34










  • $begingroup$
    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.
    $endgroup$
    – Winther
    Dec 12 '18 at 10:37












  • $begingroup$
    There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
    $endgroup$
    – Winther
    Dec 12 '18 at 10:40










  • $begingroup$
    @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.
    $endgroup$
    – Awe Kumar Jha
    Dec 12 '18 at 10:45


















$begingroup$
Possible duplicate of How to solve an nth degree polynomial equation
$endgroup$
– Winther
Dec 12 '18 at 10:20




$begingroup$
Possible duplicate of How to solve an nth degree polynomial equation
$endgroup$
– Winther
Dec 12 '18 at 10:20












$begingroup$
@Winther, this is not a duplicate, I have edited the question, please consider it again.
$endgroup$
– Awe Kumar Jha
Dec 12 '18 at 10:34




$begingroup$
@Winther, this is not a duplicate, I have edited the question, please consider it again.
$endgroup$
– Awe Kumar Jha
Dec 12 '18 at 10:34












$begingroup$
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.
$endgroup$
– Winther
Dec 12 '18 at 10:37






$begingroup$
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.
$endgroup$
– Winther
Dec 12 '18 at 10:37














$begingroup$
There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
$endgroup$
– Winther
Dec 12 '18 at 10:40




$begingroup$
There are general solutions (in non-elementary functions) for quintics. See for example How to solve fifth-degree equations by elliptic functions?
$endgroup$
– Winther
Dec 12 '18 at 10:40












$begingroup$
@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.
$endgroup$
– Awe Kumar Jha
Dec 12 '18 at 10:45






$begingroup$
@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.
$endgroup$
– Awe Kumar Jha
Dec 12 '18 at 10:45












1 Answer
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$begingroup$

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.






share|cite|improve this answer









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    1 Answer
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    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 12 '18 at 10:37









        Giuseppe NegroGiuseppe Negro

        17.1k330124




        17.1k330124






























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