information:classification
Asymptotic expansion

An asymptotic expansion of a function f (x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide an increasingly accurate description of the order of growth of f.

In symbols, it means we have f ∼ g 1, {\displaystyle f\sim g_{1},} but also f − g 1 ∼ g 2 {\displaystyle f-g_{1}\sim g_{2}} and f − g 1 − ⋯ − g k − 1 ∼ g k {\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}} for each fixed k. In view of the definition of the ∼ {\displaystyle \sim } symbol, the last equation means f − (g 1 + ⋯ + g k) = o (g k) {\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})} in the little o notation, i.e., f − (g 1 + ⋯ + g k) {\displaystyle f-(g_{1}+\cdots +g_{k})} is much smaller than g k. {\displaystyle g_{k}.}

The relation f − g 1 − ⋯ − g k − 1 ∼ g k {\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}} takes its full meaning if g k + 1 = o (g k) {\displaystyle g_{k+1}=o(g_{k})} for all k, which means the g k {\displaystyle g_{k}} form an asymptotic scale. In that case, some authors may abusively write f ∼ g 1 + ⋯ + g k {\displaystyle f\sim g_{1}+\cdots +g_{k}} to denote the statement f − (g 1 + ⋯ + g k) = o (g k). {\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).} One should however be careful that this is not a standard use of the ∼ {\displaystyle \sim } symbol, and that it does not correspond to the definition given in § Definition.

In the present situation, this relation g k = o (g k − 1) {\displaystyle g_{k}=o(g_{k-1})} actually follows from combining steps k and k −1; by subtracting f − g 1 − ⋯ − g k − 2 = g k − 1 + o (g k − 1) {\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})} from f − g 1 − ⋯ − g k − 2 − g k − 1 = g k + o (g k), {\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),} one gets g k + o (g k) = o (g k − 1), {\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),} i.e. g k = o (g k − 1). {\displaystyle g_{k}=o(g_{k-1}).}

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.