I'm trying tu calculate the eigenvalues of Legendre differential operator, that is:
$$-\frac{d}{dx}\left( (1-x^2) \frac{du}{dx}\right) = \lambda\, u \,\,\,\text{in}\,\,\,x\in[-1,1]$$
Using series of powers is easy to prove that,
$$C_1\sum_{n=0}^{\infty}a_{2n}\,x^{2n}+C_2\,\sum_{n=0}^{\infty}a_{2n+1}\,x^{2n+1}$$
are two independient solutions with
$$a_{n+2}=-a_n\frac{\lambda-n(n+1)}{(n+1)(n+2)}$$
Both series converge in $(-1,1)$. To check the convergence at $x=-1$ or $1$ we find the following series:
$$\sum_{n=0}^{\infty}a_n$$ If $\lambda=n(n+1)$, solutions are Legendre polynomials, thus convergent.
My question is how to prove that there are not any other values of $\lambda$ that makes this series convergent (...or this assertation is false !).
Any help or bibliografhic reference is welcomed.