Sometimes you need to integrate only the ``tail'' of a Gaussian function, from some large x up to infinity:
∫x∞e−t2dt=?when x≫1.
Evaluate this integral approximately as follows. First, change variables to s=t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2 in a Taylor series about that point, keeping only the first few terms in the series. Do this to obtain a series expansion for the integral. Evaluate the first three terms of the series explicitly to obtain
∫x∞e−t2dt=e−x2(2x1−4x31+8x53−⋯).
Note: When x is fairly large, the first few terms of this series will converge very rapidly toward the exact answer. However, if you calculate too many terms, the coefficients in the numerators will eventually start to grow more rapidly than the denominators and the series will diverge. This happens sooner or later no matter how large x is! Series expansions of this type are called asymptotic expansions. They're incredibly useful, though they make me rather queasy.