According to The Devil's Invention by Boyd we have
PROPOSITION 2 (Exponential Reciprocal Rule). If a function f(ε) contains a term which is an exponential function of the reciprocal of ε, then a power series in ε will not converge to f(ε).
The problem is, of course, that factors like $exp(−q/ε)$ are invisible to the usual perturbative approach: $exp(−q/ε) ∼ 0 + 0ε + 0ε^2 +\ldots$
This immediately means that if we try to approximate a function like $$ h(ε) ≡ \sqrt{1 + ε} + exp(−1/ε)$$ with a series, this series will converge for $|ε| < 1$, but to a wrong value for all $ε$ except $ε = 0$.
So far, so good.
My problem is understanding why "this situation – a convergent series for a function that contains a term exponentially small in 1/ε, and therefore invisible to the power series – seems to be rare in applications."(The Devil's Invention by Boyd)
Instead, usually a factor like $exp(−1/ε)$ leads to a divergent series. If we try to approximate it with the usual methods, we get a good approximation, but only up to some fixed order $\sim O(1/\epsilon)$.
Is there some easy example or good way to see why and how such factors that are invisible to the usual series approach can lead to a divergent series? And if yes, why does this divergence start at $\sim O(1/\epsilon)$?