Integration by parts approximation

the boundary terms can give an Asymptotic Approximation under certain circumstances

Example

as .

Need to show that
Can show as
So as

We can keep integrating by parts to find

(this diverges, but that is okay as we can Optimal Truncation)

Example

Find the behaviour of as , for .
As , the integrand is dominated by a small region near .

Repeated integration by parts:

The first sum is an term approximation of ,
with as ,
but we need to show that the other two are .

If is infinitely differentiable we have