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although it does converge because
n
! grows faster than
x
n
for any constant
x
. However,
when
x
is small, either positive or negative, the series converges rapidly, and only a few
terms are necessary to get a good approximation.
EXAMPLE
1.6 Let
x
= 1/2. Then
or approximately
e
1/2
= 1
.
64844.
Let
x
= −1. Then
or approximately
e
−
1
= 0
.
36786.
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1.3.6
Power Laws
There are many phenomena that relate two variables by a
power law
, that is, a linear rela-
tionship between the logarithms of the variables.
Figure 1.3
suggests such a relationship.
If
x
is the horizontal axis and
y
is the vertical axis, then the relationship is log
10
y
= 6 − 2
log
10
x
.
Figure 1.3
A power law with a slope of −2
EXAMPLE
1.7 We might examine topic sales at
Amazon.com
, and let
x
represent the rank of
topics by sales. Then
y
is the number of sales of the
x
th best-selling topic over some period.
The implication of the graph of
Fig. 1.3
would be that the best-selling topic sold 1,000,000
copies, the 10th best-selling topic sold 10,000 copies, the 100th best-selling topic sold 100
copies, and so on for all ranks between these numbers and beyond. The implication that
above rank 1000 the sales are a fraction of a topic is too extreme, and we would in fact
expect the line to flatten out for ranks much higher than 1000.
□
