How can an asymptotic expansion give an extremely accurate prediction, as in QED?

Suppose you're interested in computing some quantity $F(alpha)$, like the excess magnetic moment $g-2$, which depends on the fine structure constant $alpha simeq .007$.

Perturbation theory gives a recipe for the coefficients $F_i$ of an infinite series $sum_{i geq 0} F_i alpha^i$, which is expected to be asymptotic to $F$. This means that, if you add up the first few terms, you should get a decent approximation to $F$.

$F simeq F_0 + F_1 alpha + F_2 alpha^2 + F_3 alpha^3 + ... + F_N alpha^N$

If you keep adding higher order corrections, making $N$ bigger, the approximation will get better for a while, and then, eventually, it will start to get worse and diverge in the limit $N to infty$. In QED, we don't know exactly where the approximation gets worse. However, one expects on general grounds that the series approximation will start to get worse when $N simeq 1/alpha simeq 137$. (And this is indeed what we see in toy models, where we can make everything completely rigorous.)

In principle, then, we should add up all the Feynman diagrams of order $leq 137$. In practice, we don't have the ability to do this. Computing all those diagrams is very time-consuming. Even with computer assistance, we have difficulty going beyond 4 or 5 loops.