What is the minimum speed for convergence? I've read several passages that state things like 'the function doesn't decrease fast enough and hence doesn't converge'. How fast is fast enough? 
So far the only explanation I've found is basically to compare the line $y=e^{-x}$ to $y=5$. The line $y=e^{-x}$ will converge to zero (as $x$ increases to infinity) while the line $y=5$ won't. I get the idea but it doesn't tell me anything about what the 'speed' should be, or how to measure it.

A function $f(x)$ can decrease extremely slowly and still converge to zero as $x \to \infty.   f(x)=\frac 1{\log(\log(x))}$ would be an example.  You need a function to decrease quickly (faster than $\frac 1x$) if you want its integral to converge.
