Test functions as a taylor series with integral remainder

Hi All,

Given a test function (C^infinity function with compact support), I have shown that we can write

Choose a test function on such that on a neighbourhood of . I have to show the following:

On this neighbourhood, any test function on can be written as

where is a test function on such that for .

I cannot see where this is coming from.

I have considered the function defined by , but I do not think this has the desired derivative properties.

Can anyone help?