Use the Recurrent relation for generating the Chebyshev polynomials which is given by To(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) –Tn-1(x) to  approximate the function   with a Maclaurin’s polynomial of order 5. Hence use the Chebyshev polynomials and method of economization to obtain a lesser degree polynomial of approximation, of order 3. 

(a) Use the Recurrent relation for generating the Chebyshev polynomials which is given by To(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) –Tn-1(x) to  approximate the function   with a Maclaurin’s polynomial of order 5. Hence use the Chebyshev polynomials and method of economization to obtain a lesser degree polynomial of approximation, of order 3.

(b) Determine the upper error bound, due to the economization.

(c) Compare and contrast with results obtained using Legendre’s orthogonal polynomials.

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