Zero-Order Remainders

Speaking of remainders that are interesting but useless, notice that the 
series expansion of (1+x)^2 can be written as

              (2)     (2)(1)       (2)(1)(0)       (2)(1)(0)(-1)
(1+x)^2 = 1 + --- x + ------ x^2 + --------- x^3 + ------------- x^4 +
               1!       2!            3!                4!

where every term after the third has a zero in the numerator. Factoring 
the zero out of the remainder terms, we have
 
               (1+x)^2  =  (1 + 2x + x^2) +  (0) R(x)

where
                                inf    n!   
                 R(x)  =  2x^3 SUM   ------ (-x)^n
                                n=0  (n+3)! 

This converges for |x| < 1, but what, if any, significance it has
relative to (1+x)^2, I don't know.  A similar "order zero" remainder 
function can be associated with any finite expansion.