= Solution
Write $P_n(x)\sim c_nx^n$ with $c_n\ne0$. For $n\geq1$, this branch grows. Choosing the other <reduction of order> solution by an integral from infinity gives, on $x>1$,
$$
P_n(x)\int_\infty^x\frac{dt}{(1-t^2)P_n(t)^2}\sim\frac{x^{-n-1}}{c_n(2n+1)}.
$$
Thus \b[for $n\geq1$, every bounded exterior solution decays as $|x|^{-n-1}$], up to a constant factor, on each exterior interval. Equivalently the dominant-balance exponents in the <Legendre differential equation> are $n$ and $-n-1$.
There is an important exception at degree zero. Since $P_0=1$,
$$
\boxed{y=C_1+\frac{C_2}{2}\log\left|\frac{x+1}{x-1}\right|=C_1+\frac{C_2}{x}+O(|x|^{-3})\quad(n=0).}
$$
These solutions are bounded even when $C_1\ne0$; boundedness alone does not then imply decay to zero. The constants on $x>1$ and $x<-1$ are independent unless continuation across the singular endpoints is specified.
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