Let be $f:[0, \infty)\rightarrow [0,\infty)$ a measurable function such that $$\int_0^{\infty}x\cdot f(x)\,dx< \infty.$$ Show that $$\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx<\infty .$$
Progress
The only thing I proved is that $n\cdot \int_0^{\infty}f(x+n)dx\rightarrow 0$. Clearly it is not enough.
Try it with the function $f(x) = \dfrac 1x$ for $0 < x \le 1$ and $f(x) = 0$ otherwise.
If the sum starts at $n=1$, use $\displaystyle \int_0^\infty f(x+n) \, dx = \int_n^\infty f(x) \, dx = \int_0^\infty \chi_{[n,\infty)}(x) f(x) \, dx$ along with $\displaystyle \sum_{n=1}^\infty \chi_{[n,\infty)}(x) \le x$.