Calculus: Analytic Geometry and Calculus, with Vectors

12.6 Euler-Maclaurin summation formulas 649

that there are constants M and Dnx such that IDnxj 5 M and

(14) log Zn (k/ =log In - 2X2 + Dnx

This shows that if (10) and (11) hold, then

(15) eMln nae 2 ' <_2n\k)1 (nl< euln' 2 e 2as.

Vnr

Suppose now that xi and x2 are two numbers, not necessarily positiNe, for %hich

rl < x2 and suppose that a has been chosen such that Jxj < a and Jx2j < a

Let

(16) Pn(x1,x2) =

Since lira e -Min = I and lim eMln = 1, it follows from (15) that

(17)

n- -

lira P,(x,,xs) =lim

n-, n-,

. n

n- -

provided the limit on the right exists. In (17) and elsewhere, a star on a sigma

indicates that the range of summation is the same as that in (16). For present

purposes, let the number X in (6), (15), and (17) be denoted by X, , so that

k=Z+XxV.

Then )Ln.k - Xn.x_1 = 1//, and consequently

(18)^11

nIr G it

The right member of (18) is, except for negligible discrepancies at the ends of the

interval x, 5 x < X2, a Riemann sum which converges to the right member of

the formula

(19) lim* e-244- = r x: e2:=dz.

n-. m v na 7C

Therefore, (19) holds. From (19), (17), and (16) we obtain the formula

(20) I

sLsaj+XSV/

J z, `

g:= dx.

2n(k 4'r

In order to compare this result with other statistical results, we replace x, and

x2 by w,/2 and w2/2 in (20). Changing the variable of integration by setting