Calculus: Analytic Geometry and Calculus, with Vectors

12.6 Euler-Maclaurin summation formulas 643

In many practical applications, the values of the integrals

(12.67) f 2 f (m) (x)Bm(x) dx, f,' f (m) (x)B,, (x) dx

are not known, but this makes no difference because their algebraic signs
can be determined and we discover that numbers we want to calculate
lie between known numbers that are extremely close together. For
example, the natures of B3(x) and Bs(x) are such that if f(3)(x) and f(5)(x)
are positive and decreasing, then the integrals in (12.67) are positive
when m = 3 and are negative when m = S.
The first problems at the end of this section give some of the simple
applications of the Euler-Maclaurin summation formulas. In Problem
4 we shall derive some very important Stirling formulas. In Problem 5
we shall give an elementary proof of the fact that if co, < W2 and

(12.68) Pn(W1,W2) =

(^11) n!
2" k! (n - k) 1
2
then
(12.681)^1
lim Pn(Wl,W2) = x11'2 dx.
n- w 2a
fe
,
This and related formulas are very important in probability and statistics.
Persons who have or will have interest in these matters are well advised
to complete this course and proceed to study an authoritative textbook
by Fellert which many people read just for the fun of it.

Problems 12.69

1 Show that putting p = 0, q = n, and f (x) = x2 in (12.662) gives the
formula
k2_n(n-h1)(2n+1)
7c-1 6

2 Supposing that s is a positive integer, show that putting p = 0, q = n,
m = s, and f(x) = x8 in (12.663) gives the formula

k8 =

?Z8+1

+

n8

+ s(s - 1) (s - 2) ... (s - j + 2)ne-,+1$i.

k-1 s + 1^2 =2

Remark: The result can be put in the neater form

1

I ks = ns + + 1 1 1'z$+l-,B,

k-1 s+110(^9 I

t William Feller, "An Introduction to Probability Theory and its Applications," John

Wiley & Sons, Inc., New York, 1957.

n a