Page 303 - Elements of Distribution Theory

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9.7 Approximation of Sums 289

where m and r are nonnegative integers m ≤ r. Here any or all of m,r, and f may depend
on a parameter n and we consider an approximation to this sum as n →∞.
One commonly used approach to approximating this type of sum is to first approximate
the sum by an integral and then approximate the integral using one of the methods discussed
in this chapter. The basic result relating sums and integrals is known as the Euler-Maclaurin
summation formula. The following theorem gives a simple form of this result. The general
result incorporates higher derivatives of f ; see, for example, Andrews, Askey, and Roy
(1999, Appendix D) or Whittaker and Watson (1997, Chapter 13). Thus, the approximations
derived in this section tend to be rather crude; however, they illustrate the basic approach
to approximating sums.

Theorem 9.17. Let f denote a continuously differentiable function on [m,r] where m and
rare integers, m ≤ r. Then
r r r
1

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

Proof. The result clearly holds whenever m = r so assume that m < r. Let j denote an
integer in [m,r). Consider the integral

j+1

P(x) f (x) dx.
j
Note that, on the interval ( j, j + 1), P(x) = x − j − 1/2. By integration-by-parts,

j+1 j+1 j+1

− f (x) dx
P(x) f (x) dx = (x − j − 1/2) f (x)
j j j
1 j+1
= [ f ( j + 1) + f ( j)] − f (x) dx.
2 j
Hence,
r r−1 j+1

P(x) f (x) dx = P(x) f (x) dx
m j=m j
r r−1
1 r
= f ( j) + f ( j) − f (x) dx
2 m
j=m+1 j=m
r r
1
= f ( j) − [ f (m) + f (r)] − f (x) dx,
2
j=m m
proving the result.
The same approach may be used with infinite sums, provided that the sum and the terms
in the approximation converge appropriately.