Actually using a non-rigorous derivation the R doesn't even appear.I'd just like to confirm that if the summation over k goes to infinity (as it does in the above version) then we don't actually need R at all. So is it true that R is an error term that is non-zero only if we truncate the summation terms over k at some k = p?
Note here the sum is from 0 to n-1.
To make the above rigorous, should that appeal, see:
But I personally would not worry. Arguments like that are used in physics and applied math all the time - if you get too caught up in it you will find it takes up your time for no gain in using it to solve problems. But sometimes you just can't resist - I now that feeling only too well.