1. When Fourier first introduced his series in 1807, many leading mathematicians, including Poisson, Laplace and Lagrange, were skeptical. For example, by calculating some coefficients, Fourier arrived at the amazing formula, COST 1 100 An2 - 1 sin 2nz but did not write down the interval of validity. Laplace would not accept that this expansion was true because cosine is an even function, while sine is an odd function, and a sum of odd functions is odd. (a) Sketch the graph of the sum of the Fourier series in the right-hand side of the above formula. (b) Deduce the interval of validity of the above expansion, showing that Laplace's concerns were unwarranted. Repeat Fourier's calculation of the series coefficients. (c) By termwise integration, derive the cosine series for sin z, and sketch the graph of the sum of the series