Calculus: Analytic Geometry and Calculus, with Vectors

12.6 Euler-Maclaurin summation formulas 647

and more formulas can be produced by giving greater values to m. While these
formulas have important applications to cases in which z is a complex number,
we confine ourattention to the case in which z is real and z > 0. When p is a
positive integer, the function (z + x)_1' is then positive and decreasingover the
infinite interval x Z 0. Hence properties of the Bernoulli functions (those
revealed in a problem at the end of Section 5.3) enable us to show that the integrals
in (16), (17), (18), (19) are respectively negative, positive, negative, positive.
Hence

(20) E(z) > 0, E(z) <

122' E(z) > (^122) 36023'
and
(21) E(z) < 122 360-3 + 1260z5
Even when z is as small as 2 or 3, this gives remarkably precise information about
E(z). In very many applications of these things, z is a positive integer n and
(15) is put in the form
(22) n! = 2nr n"e-"een112n,
where 0 < On < 1 and 0n is near 1 whenever n is large. In fact, putting E(z)
E(n) = 0n/12n in (20) and (21) shows that
(23) 1 30n2< 0n < 1 30n2 + 105n4
Thus On is quite close to 1 even when n = 1.
5 When n is a positive integer, we can put x = y = 1 in the binomial formula
(1) (x + y)n = kIok(nn!k)!xn-kyk k40
(k) xn-kyk
to obtain the formula
(2)

4 -fn

k)=I.

To obtain more information about the terms in this sum, particularly when k

is roughly n/2, we start with the formula

(3) log 2" (k) -n log 2 + log n! - log k! - log (n -k)!.

=

The last three terms can be calculated from the Stirling formula

(4) log n! = log 2r + (n + i) log n - n + En

and the results of replacing is by k and by (n - k) in it. The error term En,

which can be approximated very closely with the aid of formula (13) ofthe

preceding problem, is about 1/12n even when is is quite small. Except when is