1 8. Verify that Ento 2i n function illustrated in Figure 6, defined by f(0) =...

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1 8. Verify that Ento 2i n function illustrated in Figure 6, defined by f(0) = 0, and etx f(x) = is the Fourier series of the 2-periodic sawtooth π 2 π 2 KIN - 0 x 2 x 2 Note that this function is not continuous. Show that nevertheless, the series converges for every x (by which we mean, as usual, that the symmetric partial sums of the series converge). In particular, the value of the series at the origin, namely 0, is the average of the values of f(x) as x approaches the origin from the left and the right. if - < x < 0, T NE if 0 < x < π. T Figure 6. The sawtooth function [Hint: Use Dirichlet's test for convergence of a series anbn.] 1 8. Verify that Ento 2i n function illustrated in Figure 6, defined by f(0) = 0, and etx f(x) = is the Fourier series of the 2-periodic sawtooth π 2 π 2 KIN - 0 x 2 x 2 Note that this function is not continuous. Show that nevertheless, the series converges for every x (by which we mean, as usual, that the symmetric partial sums of the series converge). In particular, the value of the series at the origin, namely 0, is the average of the values of f(x) as x approaches the origin from the left and the right. if - < x < 0, T NE if 0 < x < π. T Figure 6. The sawtooth function [Hint: Use Dirichlet's test for convergence of a series anbn.]

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