How to tell if series terminates (Legendre ODE)
$begingroup$
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
asked Jan 1 at 9:36
user107224user107224
463314
$endgroup$
add a comment |
$begingroup$
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
asked Jan 1 at 9:36
user107224user107224
463314
$endgroup$
$begingroup$
@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
add a comment |
$begingroup$
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
asked Jan 1 at 9:36
user107224user107224
463314
$endgroup$
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
asked Jan 1 at 9:36
user107224user107224
463314
asked Jan 1 at 9:36
user107224user107224
463314
edited Jan 1 at 13:49
user107224
asked Jan 1 at 9:36
user107224user107224
463314
asked Jan 1 at 9:36
user107224user107224
463314
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
add a comment |
$begingroup$
@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
add a comment |
1 Answer
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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.
answered Jan 1 at 15:35
SomosSomos
14.4k11236
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add a comment |
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1 Answer
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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.
answered Jan 1 at 15:35
SomosSomos
14.4k11236
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 1 at 15:35
SomosSomos
14.4k11236
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 1 at 15:35
SomosSomos
14.4k11236
$endgroup$
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.
answered Jan 1 at 15:35
SomosSomos
14.4k11236
edited Jan 2 at 1:55
answered Jan 1 at 15:35
SomosSomos
14.4k11236
answered Jan 1 at 15:35
SomosSomos
14.4k11236
answered Jan 1 at 15:35
SomosSomos
14.4k11236
14.4k11236
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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