§1. Convergent sequences and series
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If(x) - g(x)1 < rlg(x)l,
in other words f(x) - g(x) = o(g(x)), or again
(4.12)
f(x) = g(x) + o(g(x)) :
f(x) is sum of g(x) and of a function negligible with respect to 9 when x
tends to the value considered. Here, as in the case of the notation O(g(x»,
the notation o(g(x» is used to denote any function negligible with respect to
g(x). Some authors introduce indices to avoid confusing functions that are in
fact distinct 01 (g), 02 (g), etc.
For example
x 2 + X rv x 2 when Ixl --+ 00
since we have seen above that x = 0(x 2 ).
Another example: to say that a function f defined on a neighbourhood
of a point a possesses a derivative at a means that there exists a constant c
such that
(4.13)
f(a + h) = f(a) + ch + o(h) when h --+ OJ
for this relation means that, for all r > 0,
If(a + h) - f(a) - chi < rlhl,
i.e.
I
f(a + h) - f(a)
I
h
- c < r,
for Ihl sufficiently small, in other words that f possesses a derivative f'(a) = c
at a. For example
sinx = x + o(x) rv X when x --+ 0
since the ratio sin x/x tends to the derivative of the function sine for x = 0,
i.e. to cos 0 = 1.
5 - Convergent sequences: definition and examples
Let us return to the principal topic of this chapter: convergent sequences. As
we said while explaining Set Theory, a sequence of elements of a set E is a
function defined for all integers n ~ 1 (or, more generally, for all sufficiently
large n E Z) with values in Ej the value of this function for x = n might,
for example, be written f(n), but the tradition has it that it is better to
write Un and to employ the notation (un) to denote the succession of values
U1, U2, ... of the terms of the sequence. This of course does not prevent us
from using functional notation u(n) when that appears more convenient,
particular for typists, a category to which mathematicians have belonged for