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and:
f
(
x
0
−
1
/
n
)
−
z
are two asymptotically equivalent infinitesimals.
The function
f
(
)
(
)
(
)
/
(
)
(
)
n
is a
zero
of
g
n
if
f
n
g
n
is an infinitesimal, and
o
g
∈
θ
(
)
denotes the class of the zeros of
g
.If
f
g
,then
f
and
g
are said to be of the
same
asymptotic order
, written
f
,then
f
is said to be of a smaller
asymptotic order with respect to
g
, and we also write
f
=
o
g
.If
f
∈
o
(
g
)
o
g
.
The growth of functions from reals to reals can be compared by considering their
restrictions to the set of the natural numbers. It can be easily shown that polynomials
of the same degree have the same asymptotic order, while those of smaller degrees
have smaller asymptotic orders. Moreover,
log
b
(
≤
x
)
≤
o
x
(for any
b
≥
2), and
x
≤
o
a
x
(for any
a
>
1). The following rules can be also proved. If
f
(
x
)
≤
o
g
(
x
)
,then
f
(
x
)+
g
(
x
)=
o
g
(
x
)
.If
f
(
x
)
≤
o
g
(
x
)
and
h
(
x
)
is an increasing real function, then
f
(
x
)
h
(
x
)
≤
o
g
(
x
)
h
(
x
)
.If
f
(
x
)
≤
o
g
(
x
)
and
h
(
x
)
is a non-decreasing real function,
then
f
. By means of these rules (about
polynomials, logarithms, exponentials, sums, products, and compositions) it is easy
to asymptotically compare a large class of real functions.
However, manipulating infinitesimals and infinites requires one to be very care-
ful, because very often the intuition may lead to wrong consequences. Let us men-
tion, for example, a surprising phenomenon related to the following
harmonic se-
quence
:
(
h
(
x
))
≤
o
g
(
h
(
x
))
and
h
(
f
(
x
))
≤
o
h
(
g
(
x
))
1
5
,...
of course it is an infinitesimal, but it can be easily shown that the following
har-
monic series
H
n
is an infinite:
1
2
,
1
3
,
1
4
,
1
,
n
i
=
1
1
i
.
H
n
=
Ta b l e 5 . 2 2
The infinity of harmonic series (Oresme, 1323 -1382)
1
2
+
1
3
+
1
4
+
1
5
+
...
H
n
=
1
+
=
2
+(
3
+
4
)+(
5
+
...
+
8
)+
... >
1
+
1
2
+(
1
4
+
1
4
)+(
1
8
+
...
+
1
8
)+
...
=
1
+
2
+
4
+
8
+
...
1
+
=
1
2
+
1
2
+
1
2
+
...
1
+
= an infinite
There exists a constant
γ
=
0
.
577215
...
(it is unknown whether
γ
is irrational),
called Euler-Mascheroni constant such that:
H
n
=
ln
(
n
)+
γ
.
