Acceleration of Convergence. In the next problem, we show

Chapter 10, Problem 19

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Acceleration of Convergence. In the next problem, we show how it is sometimes possible to improve the speed of convergence of a Fourier series.Suppose that we wish to calculate values of the function g, whereg(x) = n=1(2n 1)1 + (2n 1)2 sin(2n 1)x. (i)It is possible to show that this series converges, albeit rather slowly. However, observe thatfor large n the terms in the series (i) are approximately equal to [sin(2n 1)x]/(2n 1)and that the latter terms are similar to those in the example in the text, Eq. (6).(a) Show thatn=1[sin(2n 1)x]/(2n 1) = (/2)f(x) 12, (ii)where f is the square wave in the example with L = 1.(b) Subtract Eq. (ii) from Eq. (i) and show thatg(x) = 2f(x) 12n=1sin(2n 1)x(2n 1)[1 + (2n 1)2]. (iii)The series (iii) converges much faster than the series (i) and thus provides a better way tocalculate values of g(x).