78
II - Convergence: Discrete variables
of a real number means that it is the limit of the sequence whose terms are
80
Xo
81
Xo + xd10
82
Xo + xd10 + x2/100
etc., in other words that
00
(6.2)
x = Xo + xd10 + x2/100 + ... = L Xn/10n.
n=O
Consider for example the number
2/15 =0.13333333333333 ... ,
according to commercial arithmetic. By the above, the right hand side is the
sum of the series 1/10 + 3/100 + 3/1000 + ... , plausibly equal to
(6.3)
1/10 + 3.10- 2 (1 + 1/10 + 1/100 + ... ),
so that it comes down to calculating the sum of the geometric series
1 + q +q2 + ...
for q = 1/10. Now the partial sums are, for q =1= 1, the numbers
1 _ qn+l
1 + q + q2 + ... + qn = _-=-_
1-q
1-q
1-q
When n increases indefinitely, qn+l tends to 0 if Iql < 1 (nO 5, example 5),
so also does qn+l/(l - q) by the more elementary rules that one will find in
nO 8. One deduces that
(6.4)
1
1 + q + q2 + ... = ~ qn = - - if Iql < 1.
L
1-q
nEl\I
In particular,
1
1 + 1/10 + 1/100 + ... = 1 _ 1/10 = 10/9.
Thus one finds the value 1/10 + 3.10- 2 .10/9 for the series (3) and it remains
to verify that this result is in fact the fraction 2/15 from which we started.
On replacing q by -q in (4), one finds the formula
(6.5)
1
2
3
_ ~( l)n n _ _ 1_
-q+q -q +···-L -
q -l+q
n=O