Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 569 — #577
14.6. Double Trouble 569
Approximation n 1 n 10 n 100 n 1000
p
2 n
n
e
n
< 10% < 1% < 0.1% < 0.01%
p
2 n
n
e
n e
1=12n
< 1% < 0.01% < 0.0001% < 0.000001%
Table 14.1 Error bounds on common approximations for nŠ from Theorem
14.5.1. For example, if n 100, then p 2n n n
e approximates nŠ to
within 0.1%.
Corollary 14.5.2.
nŠ < p n
n
8ˆ< 1:09 for n 1;
2n 1:009 for n 10;
e ˆ:
1:0009 for n 100:
14.6 Double Trouble
Sometimes we have to evaluate sums of sums, otherwise known as double summations.
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