Exercice 5 Let If.Im be the sequence of functions defined on R by f.(x) = am ,n21. 1. Prove that the series _ f.(x) is convergent for every x E R. 1=1 2. Let S(x) = > f.(x). Prove that the function S is continuous. 3. Prove that S(x)dx = 2 (2n + 1)4 1=0 (Hint : For every n E N, cos(na) = (-1)"). 4. Show that for every x e R, we get S'(x) = cos(nx) 2 5. Deduce that (-1 )" (2n + 1)3 - (Hint : For every n e N, sin(na) = 0 and sin(nx + ;) = (-1)")