This is the desired expression for the remainder in the Adams-Bashforth formula.
In a completely analogous manner, we find for the Adams-Moulton formula
where
and
The Adams-Moulton formula is obtained by neglecting the remainder term in (7.26). Again, we use the following two facts about
is of constant sign in the interval and
is a continuous function of .
Thus applying second mean value theorem of integral calculus, we have
for some satisfying . By definition of , we have
in (7.26). Again, we use the following two facts about 
is of constant sign in the interval 
and 
is a continuous function of 
.
satisfying 
. By definition of 
, we have 
