divergent asymptotic expansion

Here is one of the most delightful paradoxes in applied mathematics: a series whose terms eventually blow up — so that summing all of it gives infinity or nonsense — can still be one of the most accurate numerical tools you own. The trick is that you never add up the whole thing. You add a few terms, stop, and the partial sum hugs the true value remarkably closely. This is the everyday reality of working with asymptotic series.

How it works: in a typical asymptotic expansion the terms first decrease, reach a smallest term, and then grow without bound (often because the coefficients grow like a factorial). For a fixed value of the variable, the best you can do is truncate at or just before that smallest term — the so-called optimal truncation. The error you incur is roughly the size of the smallest term itself, which for many problems is exponentially small. Pushing past the smallest term only injects the runaway tail and ruins the answer. The mathematician George Stokes used such a divergent series to compute Airy-function values to great precision in the 1850s, long before anyone fully justified it.

This matters because the natural-feeling demand 'the series must converge to be trustworthy' is simply false for asymptotics. Quantum mechanics (perturbation series for energy levels are typically divergent), fluid dynamics (boundary-layer expansions), and special-function evaluation all lean on this. Modern refinements (Borel summation, resummation, exponential asymptotics) can even recover the exponentially small information hidden beyond optimal truncation. The honest caveat: a divergent expansion is reliable only when truncated wisely and only for variable values where the smallest term is genuinely small; blindly summing it, or using it where its smallest term is large, gives garbage.