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For example, if fn+1 ≥ fn for all n ≥ 0, then we have x−n f (x) ≥ fn + fn+1 x + fn+2 x2 + · · · ≥ fn (1 − x)−1 . 26) For fn = p(n), the partition function, then yields an upper bound for p(n) that is too large by a factor of n1/4 . 1, one should choose x ∈ (0, R) carefully. Usually there is a single best choice. In some pathological cases the optimal choice is obtained by letting x → 0+ or x → R− . However, usually we have limx→R− f (x) = ∞, and [z m ]f (z) > 0 for some m with 0 ≤ m < n as well as for some m > n.

If there is any main theme to this chapter, it is that generating functions are usually the easiest, most versatile, and most powerful way to study asymptotic behavior of sequences. Especially when the generating function is analytic, its behavior at the dominant singularities (a term that will be defined in Section 10) determines the asymptotics of the sequence. When the generating function is simple, and often even when it is not simple, the contribution of the dominant singularity can often be determined easily, although the sequence itself is complicated.

However, there is a class of them for which an old technique, the Lagrange-B¨ urmann inversion formula, works well. 62) n=0 is a formal power series with f0 = 0, f1 = 0, then there is an inverse formal power series f −1 (z) such that f (f The coefficients of f −1 −1 (z)) = f −1 (f (z)) = z . 63) (z) can be expressed explicitly in terms of the coefficients of f (z). More generally, we have the following result. Lagrange-B¨ urmann inversion formula. Suppose that f (z) is a formal power series with [z 0 ]f (z) = 0, [z 1 ]f (z) = 0, and that g(z) is any formal power series.