Exercise 4, § 3

a) Show that $\frac{\sin x}{\sqrt{x}} \sim \frac{\sin x}{\sqrt{x}} \left( 1 + \frac{\sin x}{\sqrt{x}} \right)$, but that the integral $\int_a^{+\infty} \frac{\sin t}{\sqrt{t}} \, dt$ converges while the integral $\int_a^{+\infty} \frac{\sin t}{\sqrt{t}} \left( 1 + \frac{\sin t}{\sqrt{t}} \right) dt$ is infinite.

b) Show that the integrals $\int_a^{+\infty} \frac{\sin t}{t} \, dt$ and $\int_a^{+\infty} \frac{\sin^2 t}{t^2} \, dt$ are both convergent but that $\int_a^{+\infty} \frac{\sin^2 t}{t^2} \, dt$ is not negligible relative to $\int_a^{+\infty} \frac{\sin t}{t} \, dt$, even though $\sin^2 t / t^2 \ll \sin t / t$.

c) Consider the two continuous functions with values in $\mathbf{R}^2$, defined on $[1, +\infty[$:

Show that they do not vanish for any value of $x$, that $f \sim g$ as $x$ tends to $+\infty$, that the integrals $\int_a^{+\infty} f(t) \, dt$ and $\int_a^{+\infty} g(t) \, dt$ converge, but that $\int_a^{+\infty} f(t) \, dt$ is not equivalent to $\int_a^{+\infty} g(t) \, dt$ as $x$ tends to $+\infty$.