Calculus: Analytic Geometry and Calculus, with Vectors

642 Series

which is used to estimate sums. More Euler-Maclaurin formulas

(12.661) f(k) =f f(x) dx +f(P)

2

f(q) +f (g) 12f'(P)

k=p p

- rf"(x)B2(x) dx,

(12.662) Zf(k) f 4f(x) dx -1- f(P)

2

f(g) + f'(q) 12f'(P)

p

+f'f "(x)B3(x) dx,

p

etcetera, are easily obtained by further integrations by parts. The

formula obtained after m integrations by parts is

f12

(12.663) kpf(k) = AX) dx+f(P) +f(q) 2

• I [fli-1) (q)- f(t-I)(P) + (-1)m+1

fOf(m)(x ')Bm(x) dx.

This formula reduces to (12.66), (12.661), and (12.662) when in is 1, 2,

and 3. We must observe that B, = 0 when j is odd and j > 3; otherwise,

some of the signs in (12.663) would be wrong.

In some important applications, f(-)(x) -> 0 and as x -+ oo and the

integral in

(12.664) CD f(P) - I fu-')(P)B + (-1)m+, f

°°

f(m)(x)Bm(x) dx

=2 j p

exists when in is sufficiently great, say m >- mo > 1. In such cases,

we can define the constant Cp by the formula (12.664), the right member

being independent of m because integration by parts shows that it is

unchanged when m is replaced by m + 1. Subtracting (12.664) from

(12.663) gives the formula

(12.665) I f(k)= CP +f 4f(x) dx +f(2 q)

E

f f(m) (x)Bm(x) dx.

Solving this formula for Cp gives the formula

(12.666) Cp =

f(k) -f 4f(x)dx

-f(2)

(q)B'

j-2

(-1)m 1 f(m)(x)Bm(x) dx,

4

which is sometimes used to calculate approximationsto Cp.