suppose $f$ is differentialbe on $[0,\infty)$, and $f'(x)\geq 0$, $f(0)>0$. Furthermore, $$\int_0^ \infty \frac{dx}{f(x)+f'(x)}<\infty,$$ show that $$\int_0^\infty \frac{dx}{f(x)}<\infty.$$
this troubles me for several days. My idea is $e^x(f(x)+f'(x))=(e^xf(x))'$.
\begin{align*} \int_{0}^{+\infty}\frac{1}{f(x)}{\rm d}x-\int_{0}^{+\infty}\frac{1}{f(x)+f'(x)}{\rm d}x&\leq\int_{0}^{+\infty}\frac{f'(x)}{f^2(x)}{\rm d}x\\ &\left.-\frac{1}{f(x)}\right|_{0}^{+\infty}\leq\frac{1}{f(0)} \end{align*} Thus, $\int_{0}^{+\infty}\frac{1}{f(x)}{\rm d}x$ converges.