 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/8/a/solution

Insert the power series and equate the coefficient of

x^{n}

(n + 2) (n + 1) a_{n + 2} + [ℓ (ℓ + 1) - n (n + 1)] a_{n} = 0.

a_{n + 2} = \frac{( n - ℓ ) ( n + ℓ + 1 )}{( n + 1 ) ( n + 2 )} a_{n}, n \geq 0.

a_{0}, a_{1}

generate the independent even and odd series, respectively. They converge at least for

∣ x ∣ < 1

, up to the nearest singular points of the equation.

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