# How to tell if series terminates (Legendre ODE)

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!

• @Somos sorry, I don’t really understand what you’re trying to get at Jan 1, 2019 at 13:48
• @Somos zero, but this still doesn’t give me clear definitions of terminated vs infinite Jan 1, 2019 at 14:53
• @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? Jan 1, 2019 at 15:20

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.