asymptotic series
An asymptotic series is a sequence of correction terms that describes a function ever more accurately near a particular point or limit — but improves only up to a sweet spot, after which adding more terms makes things worse. It trades the comfortable idea of a convergent power series for a different and very practical promise: stop at the right place and the truncated sum is spectacularly close to the truth.
The precise (Poincare) meaning: a series a_0 + a_1/x + a_2/x^2 + ... is an asymptotic expansion of f(x) as x tends to infinity if, for each fixed number of terms N, the error after the first N terms is little-o of the last term kept — that is, the remainder shrinks faster than 1/x^N. The key word is 'each fixed N': we fix the number of terms and let x grow, never the reverse. This is the opposite of how a convergent series works, where you fix x and let the number of terms grow. Because of this swapped order of limits, an asymptotic series can be wildly divergent as an infinite object yet superbly accurate when chopped off.
These expansions are the working currency of mathematical physics and applied mathematics: the large-x behaviour of the error function, Bessel functions, the exponential integral, and countless special functions are known only as asymptotic series. In practice one uses the rule of thumb 'truncate just before the smallest term', which gives the best accuracy a given expansion can offer. The honest misconception to kill: an asymptotic series need not converge, and convergence is not the goal — controlled, bounded error for accessible values of the variable is.
For large x the exponential integral gives e^x E_1(x) ~ 1/x - 1/x^2 + 2/x^3 - 6/x^4 + ...; the n-th term grows like n!/x^(n+1), so the series diverges, yet a few terms at x = 10 are very accurate.
The factorial growth of the coefficients means the infinite series diverges; truncating near the smallest term gives the best possible answer.
Two different functions can share the very same asymptotic series (they differ by something exponentially small, like e^(-x), which is o of every power of 1/x). So an asymptotic series does not, by itself, determine the function uniquely.