(5) Consider the step function 1 if@ € (0, т], O if 0 0, f(0) = if 0 € (-T,0) -1 (a) Use Dini's criterion to show that the Fourier series of f converges to f pointwise on T (b) Show that sin((2k1)0) (2k1 SNf(0) k 0 Conclude that the alternating harmonic series over odd integers sums to ; that is, Σ T 2k1 k 0 4 (c) Prove that 2 (sin(2(N1)0) sin(0) (SNf(0) TT Conclude that lim (SNf'(0)= 0. N-o0 HINT: Use the formula for the N-th partial sum of the geometric series: N-1 1 - zN 1 z k-0

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