= Solution
Differentiate the <power series> term by term inside its <radius of convergence>, and reindex the second <derivative> so that every coefficient multiplies $x^m$. The <differential equation> becomes
$$
\sum_{m=0}^\infty\left[(m+2)(m+1)a_{m+2}+
\{n(n+1)-m(m+1)\}a_m\right]x^m=0.
$$
The uniqueness of power-series coefficients gives
$$
\boxed{a_{m+2}=\frac{m(m+1)-n(n+1)}{(m+1)(m+2)}a_m
=\frac{(m-n)(m+n+1)}{(m+1)(m+2)}a_m,\quad m\geq0.}
$$
The even coefficients depend only on $a_0$ and the odd coefficients only on $a_1$. These are the two local solution branches of the <Legendre differential equation> around its <ordinary point> $x=0$.
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