66
II - Convergence: Discrete variables
since p! 2:: 2!(p - 2)!, whence the inequality
(4.9)
I(x + h)[n] - x[n] - hx[n-1]1 :::; Ihl[2] (Ixl + Ihl)[n-2]
similarly to (6), or
I (x + h)~] -- x[n] _ x[n-1] I :::; I~I (Ixl + Ihl)[n-1] :::; Mlhl
for, say, Ihl :::; 1. We then get (8) by letting h tend to o. More generally,
(4.10)
I (x + h)[n] - x[n] - hx[n-1] - ... - h[p]x[n-p] I
:::; Ihl[p+1](lxl + Ihl)[n- p -1] .
The concept of a limit now allows us to define the comparison relations
f(x) rv g(x),
f(x) = o(g(x))
which we abandoned to their fate at the end of the preceding nO. The first
- one says that the functions f(x) and g(x) are equivalent at infinity or on
a neighbourhood of the point a - means that the ratio f(x)lg(x) tends to 1
when x tends to the limit value considered. The second means that this same
ratio tends to 0; one says that f(x) is negligible with respect to g(x) in these
circumstances.
The second relation means that for all r > 0
If(x)1 < rlg(x)1
for x large (or for x close to a), i.e. that there exists an r' > 0 such that
(4.11)
Ix - al < r' ===} If(x)1 < rlg(x)l·
For example,
x 2 = o(x) when x --> 0
since Ixl < r implies Ix 2 1 < rlxl; so for example Ix 2 1 < 1O- 1ooo lxl once
Ixl < 10- 1000 . In C, one has similarly
x = o(x 2 ) when :t --> 00
since the relation Ixl < rlx 2 1 is satisfied once Ixl > 1/r.
As to the relation f rv g, it reduces to the one we have just described.
One can express this as: for all r > 0, one has
If(x)lg(x) - 11 < r
for x large (or close to a). But this can be written