Explanation

The usual interpretation of the statement that the n-th-order approximation of a function f around the point 0 is zero, is that, for all x in a small interval around 0, we have

|f(x)| |xⁿ⁺¹|.

Now, working with x -> x^{n + 1} as a negligible function, i.e., indistinguishable from 0, corresponds to computing mod X^{n + 1}.

In particular for polynomial functions, the n-th-order approximation is found by leaving out the terms of degree greater than n.

But even for more general functions, computing with their Taylor polynomials of degree n boils down to computations in R[X] modulo X^{n + 1}, and so can be viewed as operations in R[X]/(X^{n + 1}).