 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/20/b/solution

v

has leading term

x^{2 l}

. Its

l

th derivative has leading term

(2 l)! x^{l} / l!

, and all other terms have lower degree. Thus the Rodrigues formula for Legendre polynomials defines a polynomial with nonzero leading coefficient

P_{l} (x) = \frac{( 2 l )!}{2 ^{l} ( l ! ) ^{2}} x^{l} + lower-degree terms .

It therefore has degree exactly

l

, including the constant case

P_{0} = 1

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