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!

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.