Solution
= Solution
Let $Q_k=P^{(k)}$. Differentiating <Legendre's differential equation> $k$ times and using the <Leibniz rule> gives
$$
(1-x^2)Q_k''-2(k+1)xQ_k'
+\bigl[\lambda-k(k+1)\bigr]Q_k=0.
$$
The induction step follows by differentiating this equation once: the derivative of the coefficient $-2(k+1)x$ contributes the additional term $-2(k+1)Q_k'$. Thus
$$
\boxed{\lambda_k=\lambda-k(k+1)}.
$$
For fixed $\lambda$, this is negative for every sufficiently large $k$.