In this exercise Romberg rules are derived. Assume that ( ) is an integration rule that uses  subintervals and  is the corresponding width of each subinterval. (a) It is known that the error term for...

In this exercise Romberg rules are derived. Assume that
() is an integration rule that uses
 subintervals and
 is the corresponding width of each subinterval.

(a) It is known that the error term for the composite Simpson’s rule involves even powers of
. In particular,

Show that the integration rule

has an error that is
(
⁶).

(b) Suppose

Show that the integration rule

has an error that is
(
⁴).

(c) Suppose

Show that the integration rule

has an error that is
(
⁴).