How to tell if series terminates (Legendre ODE)












0












$begingroup$


when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation



$$a_{k+2}=frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$



What I do not understand is the distinction between infinite series and terminating series. I know that for both $k$ and $l$ even/odd the series “terminates” to the Legendre polynomials, while other combinations result in an “infinite series”.



I do not get the meanings of the quoted terms here. For one, isn’t part of the solution the sum of $c_l P_l$, which ranges from $l=0$ to $l=infty$, making that an infinite series? Likewise, doesn’t the other solution sum up nicely into a natural logarithm?



Additionally, how do I extend this concept to any other type of ODE?



Apologies for the buffoonish question here, I am incredibly confused!






ordinary-differential-equations power-series divergent-series legendre-polynomials







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  • $begingroup$
    @Somos sorry, I don’t really understand what you’re trying to get at
    $endgroup$
    – user107224
    Jan 1 at 13:48










  • $begingroup$
    @Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
    $endgroup$
    – user107224
    Jan 1 at 14:53










  • $begingroup$
    @Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
    $endgroup$
    – user107224
    Jan 1 at 15:20
















0












$begingroup$


when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation



$$a_{k+2}=frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$



What I do not understand is the distinction between infinite series and terminating series. I know that for both $k$ and $l$ even/odd the series “terminates” to the Legendre polynomials, while other combinations result in an “infinite series”.



I do not get the meanings of the quoted terms here. For one, isn’t part of the solution the sum of $c_l P_l$, which ranges from $l=0$ to $l=infty$, making that an infinite series? Likewise, doesn’t the other solution sum up nicely into a natural logarithm?



Additionally, how do I extend this concept to any other type of ODE?



Apologies for the buffoonish question here, I am incredibly confused!






ordinary-differential-equations power-series divergent-series legendre-polynomials







share|cite|improve this question











$endgroup$












  • $begingroup$
    @Somos sorry, I don’t really understand what you’re trying to get at
    $endgroup$
    – user107224
    Jan 1 at 13:48










  • $begingroup$
    @Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
    $endgroup$
    – user107224
    Jan 1 at 14:53










  • $begingroup$
    @Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
    $endgroup$
    – user107224
    Jan 1 at 15:20














0












0








0





$begingroup$


when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation



$$a_{k+2}=frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$



What I do not understand is the distinction between infinite series and terminating series. I know that for both $k$ and $l$ even/odd the series “terminates” to the Legendre polynomials, while other combinations result in an “infinite series”.



I do not get the meanings of the quoted terms here. For one, isn’t part of the solution the sum of $c_l P_l$, which ranges from $l=0$ to $l=infty$, making that an infinite series? Likewise, doesn’t the other solution sum up nicely into a natural logarithm?



Additionally, how do I extend this concept to any other type of ODE?



Apologies for the buffoonish question here, I am incredibly confused!






ordinary-differential-equations power-series divergent-series legendre-polynomials







share|cite|improve this question











$endgroup$




when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation



$$a_{k+2}=frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$



What I do not understand is the distinction between infinite series and terminating series. I know that for both $k$ and $l$ even/odd the series “terminates” to the Legendre polynomials, while other combinations result in an “infinite series”.



I do not get the meanings of the quoted terms here. For one, isn’t part of the solution the sum of $c_l P_l$, which ranges from $l=0$ to $l=infty$, making that an infinite series? Likewise, doesn’t the other solution sum up nicely into a natural logarithm?



Additionally, how do I extend this concept to any other type of ODE?



Apologies for the buffoonish question here, I am incredibly confused!






ordinary-differential-equations power-series divergent-series legendre-polynomials




ordinary-differential-equations power-series divergent-series legendre-polynomials






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 1 at 13:49







user107224

















asked Jan 1 at 9:36









user107224user107224

463314




463314












  • $begingroup$
    @Somos sorry, I don’t really understand what you’re trying to get at
    $endgroup$
    – user107224
    Jan 1 at 13:48










  • $begingroup$
    @Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
    $endgroup$
    – user107224
    Jan 1 at 14:53










  • $begingroup$
    @Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
    $endgroup$
    – user107224
    Jan 1 at 15:20


















  • $begingroup$
    @Somos sorry, I don’t really understand what you’re trying to get at
    $endgroup$
    – user107224
    Jan 1 at 13:48










  • $begingroup$
    @Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
    $endgroup$
    – user107224
    Jan 1 at 14:53










  • $begingroup$
    @Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
    $endgroup$
    – user107224
    Jan 1 at 15:20
















$begingroup$
@Somos sorry, I don’t really understand what you’re trying to get at
$endgroup$
– user107224
Jan 1 at 13:48




$begingroup$
@Somos sorry, I don’t really understand what you’re trying to get at
$endgroup$
– user107224
Jan 1 at 13:48












$begingroup$
@Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
$endgroup$
– user107224
Jan 1 at 14:53




$begingroup$
@Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite
$endgroup$
– user107224
Jan 1 at 14:53












$begingroup$
@Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
$endgroup$
– user107224
Jan 1 at 15:20




$begingroup$
@Somos doesn’t the same thing happen for all combinations of even and odd k and l? I don’t see his setting $k=l$ leads to the polynomial series too. Additionally, this seems to be specific to the Legendre equation, what about for a general ODE?
$endgroup$
– user107224
Jan 1 at 15:20










1 Answer
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If the terms of an infinite series $,s_1 + s_2 + ... + s_n + ...,$ are such that they are equal to zero after $,s_n,$, then it is said to terminate and its sum is $,s_1 + s_2 + ... + s_n,$ which is a finite sum and the series converges to it.



In the common case of a power series $,a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + ...,$ the same thing applies and a terminating power series is $,a_0 + a_1 x + a_2 x^2 + ... + a_nx^n,$ which is a polynomial and which has infinite radius of convergence.



You can look at MSE question 2573694 "Validity of terminating series solution of differential equation" for a similar situation.



Note that I think that the terminology is not a good one, but it is commonly used -- likely because of a lack of a better one.






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    $begingroup$

    If the terms of an infinite series $,s_1 + s_2 + ... + s_n + ...,$ are such that they are equal to zero after $,s_n,$, then it is said to terminate and its sum is $,s_1 + s_2 + ... + s_n,$ which is a finite sum and the series converges to it.



    In the common case of a power series $,a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + ...,$ the same thing applies and a terminating power series is $,a_0 + a_1 x + a_2 x^2 + ... + a_nx^n,$ which is a polynomial and which has infinite radius of convergence.



    You can look at MSE question 2573694 "Validity of terminating series solution of differential equation" for a similar situation.



    Note that I think that the terminology is not a good one, but it is commonly used -- likely because of a lack of a better one.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      If the terms of an infinite series $,s_1 + s_2 + ... + s_n + ...,$ are such that they are equal to zero after $,s_n,$, then it is said to terminate and its sum is $,s_1 + s_2 + ... + s_n,$ which is a finite sum and the series converges to it.



      In the common case of a power series $,a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + ...,$ the same thing applies and a terminating power series is $,a_0 + a_1 x + a_2 x^2 + ... + a_nx^n,$ which is a polynomial and which has infinite radius of convergence.



      You can look at MSE question 2573694 "Validity of terminating series solution of differential equation" for a similar situation.



      Note that I think that the terminology is not a good one, but it is commonly used -- likely because of a lack of a better one.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        If the terms of an infinite series $,s_1 + s_2 + ... + s_n + ...,$ are such that they are equal to zero after $,s_n,$, then it is said to terminate and its sum is $,s_1 + s_2 + ... + s_n,$ which is a finite sum and the series converges to it.



        In the common case of a power series $,a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + ...,$ the same thing applies and a terminating power series is $,a_0 + a_1 x + a_2 x^2 + ... + a_nx^n,$ which is a polynomial and which has infinite radius of convergence.



        You can look at MSE question 2573694 "Validity of terminating series solution of differential equation" for a similar situation.



        Note that I think that the terminology is not a good one, but it is commonly used -- likely because of a lack of a better one.






        share|cite|improve this answer











        $endgroup$



        If the terms of an infinite series $,s_1 + s_2 + ... + s_n + ...,$ are such that they are equal to zero after $,s_n,$, then it is said to terminate and its sum is $,s_1 + s_2 + ... + s_n,$ which is a finite sum and the series converges to it.



        In the common case of a power series $,a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + ...,$ the same thing applies and a terminating power series is $,a_0 + a_1 x + a_2 x^2 + ... + a_nx^n,$ which is a polynomial and which has infinite radius of convergence.



        You can look at MSE question 2573694 "Validity of terminating series solution of differential equation" for a similar situation.



        Note that I think that the terminology is not a good one, but it is commonly used -- likely because of a lack of a better one.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 2 at 1:55

























        answered Jan 1 at 15:35









        SomosSomos

        14.4k11236




        14.4k11236






























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