Page 279 - Elements of Distribution Theory

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

9.3 Asymptotic Expansions 265

m.Itis often sufficient to be able to approximate f (n) with error O(n −m ) for some given
fixed value of m, often m = 1or2.

Example 9.1. Consider the function defined by

1

∞
f (x) = exp(−t) dt, 0 < x < ∞.
0 1 + t/x
Recall that, for z = 1,
1 − z m
m−1
2
1 + z + z +· · · + z = .
1 − z
Hence,
1 t t 2 m−1 t m−1 m t m 1
= 1 − + +· · · + (−1) + (−1)
m
1 + t/x x x 2 x m−1 x 1 + t/x
so that
m−1
j j!
f (x) = (−1) + R m (x)
x j
j=0
where
m
∞
t 1 m!
|R m (x)|= m exp(−t) dt ≤ m .
0 x 1 + t/x x
It follows that
∞
j j
(−1) j!/x
j=0
is a valid asymptotic expansion of f (x)as x →∞. Thus, we may write

1 1 −2
∞
exp(−t) dt = 1 − + O(x ),
1 + t/x x
0
∞

1 1 2 −3
exp(−t) dt = 1 − + + O(x ),
1 + t/x x x 2
0
and so on. For x > 0, let
1 1 2
ˆ ˆ
f (x) = 1 − and f (x) = 1 − + .
3
2
x x x 2
ˆ
ˆ
Table 9.1 gives the values of f (x) and f (x), together with f (x), for several values of x.
3
2
Although both approximations are inaccurate for small values of x, both are nearly exact
for x = 30.
Note, however, that
∞ ∞
1
j j
(−1) j!/x = exp(−t) dt;
j=0 0 1 + t/x
in fact, the series diverges for any value of x.