This problem is hard. Consider the function f(x)=sin(n arcsin x) for x ...

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This problem is hard. Consider the function f(x)=sin(n arcsin x) for x <1. (a) Find the first and second derivatives of the function f(x). (b) Hence show that (1 − x²)ƒ''(x) − xf'(x)+n²f(x)=0. (c) Use mathematical induction to prove that for k=0,1,2,... (1-x²) f(k+²)(x)-(2k +1)xf(k+¹) (x) + (n² -k²)f(x)=0 where f)(x) denotes the kth derivative of f(x). (d) Hence derive the Maclaurin series for f(x). (e) Deduce that if n is odd, the function f(x) is a polynomial of degree n. sin(n arcsin.x) (f) Using your answer in (d) or otherwise, find lim x-0 X This problem is hard. Consider the function f(x)=sin(n arcsin x) for x <1. (a) Find the first and second derivatives of the function f(x). (b) Hence show that (1 − x²)ƒ''(x) − xf'(x)+n²f(x)=0. (c) Use mathematical induction to prove that for k=0,1,2,... (1-x²) f(k+²)(x)-(2k +1)xf(k+¹) (x) + (n² -k²)f(x)=0 where f)(x) denotes the kth derivative of f(x). (d) Hence derive the Maclaurin series for f(x). (e) Deduce that if n is odd, the function f(x) is a polynomial of degree n. sin(n arcsin.x) (f) Using your answer in (d) or otherwise, find lim x-0 X

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