With the notation as in Exercise 17, the Chartier-Dirichlet Test asserts that if f E
Chapter 10, Problem 18(choose chapter or problem)
With the notation as in Exercise 17, the Chartier-Dirichlet Test asserts that if f E 'R.*[a,y]for all y ::: a, if F(x) := J: f . is bounded on [a, 00), and if cpis monotone and x1~iomo qJ(x) = 0,then fcp E 'R.*[a, 00]. .(a) Show that the integral Jooo(1Ix) sin x dx converges.(b) Show that Jt(1 I 10x) sin x dx converges.(c) Show that Jooo(1 I .JX) cos x dx converges.(d) Show that the Chartier-Dirichlet Test does not apply to establish the convergence ofJooo(xl(x + 1 sin(x2) dx by taking f(x) := sin(x2).
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