96
II - Convergence: Discrete variables
Note that
u'v' - uv = (u' - u)(v' - v) + v(u' - u) + u(v' - v).
The two first inequalities (1) thus imply
lu'v' - uvl < r'2 + ar' where a = lui + Ivl.
For r' < 1, the right hand side is < (1 + a )r'. The first will thus be < r so
long as one chooses r' < min(l,r/(1 +a)), qed.
To deduce from this the case of Theorem 1 which interests us here, it is
enough to replace u' and v' by Un and vn; the two conditions of (1) are then
satisfied for all sufficiently large n, and the third relation provides the result.
(Here as always, one exploits the fact that if two relations are separately valid
for n large, then they are also valid simultaneously.)
Case of a quotient. Given that un/vn = Un X l/vn, it is enough to examine
l/vn and then to apply the result for a product.
The fact that Vn =F 0 for large n is clear: for n large, one has for example
IVn -vi < Ivl/2 since the right hand side is> 0; it follows that Ivnl > Ivl/2 > O.
To imitate the preceding argument let us put Vn = v'. We need to evaluate
II/v' - l/vl = lv' - vl/lvv'l·
For n large, the numerator of the right hand side is < r'. Now we have just
seen that the denominator is greater than IvI 2 /2. The right hand side is thus
< 2r' /lvI2, so < r provided that r' < rlvI 2 /2, qed.
Theorem 1 allows one to calculate a large number of limits easily, if only
very simple ones. For example, the sequence with general term
n 2 - 1
1 - l/n 2
Wn =
=
3n 2 + n + 1 3 + l/n + l/n 2
tends to 1/3 since l/n and 1/n 2 tend to 0, so that the two parts of the
fraction tend to 1 and 3 respectively.
More generally, let
f(x)
apxP + ap_lx p - 1 + ... + ao,
g(x)
bpx P + bp_1x P - 1 + ... + bo
be two polynomials of the same degree p [when one says that f is of degree
p, this means that ap =F OJ. Then 26
26 The relation below persists, with the same proof, if, in f(x)/g(x), one lets x tend
to infinity through not necessarily integer values. We mainly confine ourselves in
this chapter to limits in which the independent variable takes "discrete" values.
The next chapter will show that many of these results extend to the case of
"continuous" variables (in the sense where, in physics, one speaks of the "discrete
spectrum" and of the "continuous spectrum" of a luminous source: the first is
composed of isolated "rays" of zero width, the second of luminous "bands" of
nonzero width).
II - Convergence: Discrete variables
Note that
u'v' - uv = (u' - u)(v' - v) + v(u' - u) + u(v' - v).
The two first inequalities (1) thus imply
lu'v' - uvl < r'2 + ar' where a = lui + Ivl.
For r' < 1, the right hand side is < (1 + a )r'. The first will thus be < r so
long as one chooses r' < min(l,r/(1 +a)), qed.
To deduce from this the case of Theorem 1 which interests us here, it is
enough to replace u' and v' by Un and vn; the two conditions of (1) are then
satisfied for all sufficiently large n, and the third relation provides the result.
(Here as always, one exploits the fact that if two relations are separately valid
for n large, then they are also valid simultaneously.)
Case of a quotient. Given that un/vn = Un X l/vn, it is enough to examine
l/vn and then to apply the result for a product.
The fact that Vn =F 0 for large n is clear: for n large, one has for example
IVn -vi < Ivl/2 since the right hand side is> 0; it follows that Ivnl > Ivl/2 > O.
To imitate the preceding argument let us put Vn = v'. We need to evaluate
II/v' - l/vl = lv' - vl/lvv'l·
For n large, the numerator of the right hand side is < r'. Now we have just
seen that the denominator is greater than IvI 2 /2. The right hand side is thus
< 2r' /lvI2, so < r provided that r' < rlvI 2 /2, qed.
Theorem 1 allows one to calculate a large number of limits easily, if only
very simple ones. For example, the sequence with general term
n 2 - 1
1 - l/n 2
Wn =
=
3n 2 + n + 1 3 + l/n + l/n 2
tends to 1/3 since l/n and 1/n 2 tend to 0, so that the two parts of the
fraction tend to 1 and 3 respectively.
More generally, let
f(x)
apxP + ap_lx p - 1 + ... + ao,
g(x)
bpx P + bp_1x P - 1 + ... + bo
be two polynomials of the same degree p [when one says that f is of degree
p, this means that ap =F OJ. Then 26
26 The relation below persists, with the same proof, if, in f(x)/g(x), one lets x tend
to infinity through not necessarily integer values. We mainly confine ourselves in
this chapter to limits in which the independent variable takes "discrete" values.
The next chapter will show that many of these results extend to the case of
"continuous" variables (in the sense where, in physics, one speaks of the "discrete
spectrum" and of the "continuous spectrum" of a luminous source: the first is
composed of isolated "rays" of zero width, the second of luminous "bands" of
nonzero width).