13.6. Double Trouble 525
Approximation n1 n10 n100 n 1000
p
2n
n
e
n
<10% <1% <0.1% <0.01%
p
2n
n
e
n
e1=12n <1% <0.01% <0.0001% <0.000001%
Table 13.1 Error bounds on common approximations for nŠ from Theo-
rem 13.5.1. For example, ifn 100 , then
p
2n
n
e
n
approximatesnŠ to
within 0.1%.
Corollary 13.5.2.
nŠ <
p
2n
n
e
n
8
ˆ<
ˆ:
1:09 forn1;
1:009 forn10;
1:0009 forn100:
13.6 Double Trouble
Sometimes we have to evaluate sums of sums, otherwise known asdouble summa-
tions. This sounds hairy, and sometimes it is. But usually, it is straightforward—
you just evaluate the inner sum, replace it with a closed form, and then evaluate the
