# Fourier series

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Verify that (1/2i)sum {(e^(inx))/n} (where n does not equal zero in the sum) is the Fourier series of the 2 Pi periodic function defined by f(0)=0 and

f(x)=-Pi/2-x/2 for between -pi and 0 and f(x)= Pi/2-x/2 for x between 0 and Pi. Show that the Fourier series is convergent.

(Note that the 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.)

Hint: Use Dirichlet's test for convergence of a series

sum (a_n b_n) hide problem

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Show that the Fourier series is convergent.

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