I have a few questions from my h.w, I hope someone can help me.
the question is:
$f:[a,\infty) \rightarrow \mathbb{R}$ is a continuous and periodic function, with period of $T>0$ .
$g:[a,\infty) \rightarrow \mathbb{R}$ is a monotonic function and $\lim \limits_{x\rightarrow\infty} g(x) = 0$
Assume that $\displaystyle \int_{a}^{a+T} f(x) dx = 0$, and that $g$ is differentiable and its derivative is continuous. Prove that $\displaystyle\int_{a}^{\infty} f(x)g(x) dx$ converges.
I hope the question is clear. Thank you
Hint: (many details need to be added) $$ F(x)=\int_a^xf(x)\,\mathrm{d}x\tag{1} $$ Show that $F$ is periodic with period $T$. Furthermore, since $g$ is monotonic, $$ \begin{align} \int_a^\infty|g'(x)|\,\mathrm{d}x &=\left|\int_a^\infty g'(x)\,\mathrm{d}x\right|\\[6pt] &=|g(a)|\tag{2} \end{align} $$ Finally, $$ \begin{align} \int_a^bf(x)g(x)\,\mathrm{d}x &=\int_a^bg(x)\,\mathrm{d}F(x)\\ &=g(b)F(b)-g(a)F(a)-\int_a^bF(x)g'(x)\,\mathrm{d}x\\ &=g(b)F(b)-\int_a^b F(x)g'(x)\,\mathrm{d}x\tag{3} \end{align} $$ Show that $(3)$ has a limit as $b\to\infty$.