Page 304 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

290 Approximation of Integrals

Corollary 9.2. Let f denote a continuously differentiable function on [m, ∞] where m is
an integer. Assume that
∞ ∞

| f ( j)| < ∞, | f (x)| dx < ∞
j=m m
and
∞

| f (x)| dx < ∞.
m
Then
∞ ∞ ∞
1

f ( j) = f (x) dx + f (m) + P(x) f (x) dx
j=m m 2 m
where
1
P(x) = x − x − .
2

Proof. By Theorem 9.17, for any r = m, m + 1,...,
r r r
1

f ( j) = f (x) dx + [ f (m) + f (r)] + P(x) f (x) dx. (9.8)
j=m m 2 m
Note that, under the conditions of the corollary,
r

|P(x)|| f (x)| dx < ∞
m
and
lim f (r) = 0.
r→∞
The result now follows from taking limits in (9.8).

Using Theorem 9.17, the sum
r

f ( j)
j=m
may be approximated by the integral
r
f (x) dx
m
with error
1 r

[ f (m) + f (r)] + P(x) f (x) dx
2 m
or it may be approximated by
r
1
f (x) dx + [ f (m) + f (r)]
m 2