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.

There is no minimum speed since, for every function $f$ such that $f(x)\to0$ there exists another function $g$ much larger in the sense that $\dfrac{f(x)}{g(x)}\to0$, such that $g(x)\to0$ as well.
Likewise, for every function $f$ whose integral converges at $+\infty$, there exists another function $g$ much larger in the sense that $\dfrac{f(x)}{g(x)}\to0$, whose integral also converges at $+\infty$.
Likewise, for every convergent series $\sum\limits_nx_n$, there exists another convergent series $\sum\limits_ny_n$ whose entries are much larger in the sense that $\dfrac{x_n}{y_n}\to0$ when $n\to\infty$.
