Asymptotic Expansions

function to approximate

x!	SinIntegral[x]

number of terms

1

2

3

4

An asymptotic expansion of a function is an expression of that function as an infinite series of polynomial terms. A partial sum of that series is a polynomial that approximates the function with the approximation generally improving as more terms are used. For these two examples, as the number of terms increases, the error near

x=0

increases, yet the error for larger values of

x

decreases.