Function is monotonically decreasing from finite value to zero. When is it's integral finite?

Question

I have a "nice" (i.e. $C^{\infty}$, see pic for qualitative behaviour) function $f(r)$ , that is monotonically decreasing from a finite value at $r =0$ to zero at $r = \infty$ . Under what conditions can I conclude that $$ \int_0^\infty f(r) dr < \infty$$ ?

For what it's worth I've attached a few variations of it below, the explicit analytic form of it is rather cumbersome.

Answer 1

If you can show that :

$$f(x) =_\infty O(x^{-a}), a>1$$

You will have the convergence of the improper integral.

Answer 2

I think you can use here Cauchy- Integral Test for convergence. The test says if f is Monotonically decreasing positive function on $[0, \infty) $ then $\int_{0}^{\infty} f(x) dx$ converges if & only if $\sum_{n=0}^{\infty}$ f(n) converges.

Here, if Your function is postive on $[0, \infty)$ & the series $\sum_{n=0}^{\infty}f(n)$ converges then given integral converges.