76
II - Convergence: Discrete variables
1 = IOglO(10) = 2 54 log 10 (1 + h) rv 2 54 Mh.
To calculate loglO 2 for example, he again extracted 54 square roots of 2, and
found
1.000000000000000038477397965583 10 = 1 + h'
and since
finally he found
10glO 2 = h' /h = 0.30102 99956 63881 2.
This done, you start again with 3, 5, 7, etc. One may well suspect that, as
we shall see, more economical procedures were invented later.
We shall return to all this d propos the logarithmic functions.
Example 8. We have
(5.9)
limnl/n = 1.
Let us put nl/n = 1 + Xn where clearly Xn > O. The binomial formula
n = (1 + xn)n = 1 + n'Xn + ~n(n -1)x~ + ... ,
in which all the terms are> 0, shows that x; < 2/(n-1), an expression which
tends to O. For all r > 0, one thus has x; < r2 for n large, so 0 < Xn < r,
and thus Xn tends to 0, qed. (A sequence which tends to 0 is < r2 or r 624 for
n large since r2 > 0).
6 - The language of series
Though not basically different from that of sequences, the language of series is
frequently convenient, principally because many individual functions can be
represented more naturally by series than by sequences; their theory scarcely
appeared before Cauchy's time, the 1820s, while they are ubiquitous after
1650.
The fundamental problem is to give a meaning to a sum of an infinite
number of terms, imposing, of course, reasonable conditions; we do not propose to explain what the sum 1 -2 + 3 - 4 + 5 - ... might signify, let alone
the sum of all the real numbers 15 , for it is prudent to restrict oneself to sums
having no more than a countable infinity of terms. The terms of such a sum
may, by hypothesis, be put in the form of a sequence Ul, U2, .... Most often,
they are given in advance in this form, but there are also cases where it would
be quite artificial to order the terms of the sum as a sequence. Consider for
example the "lattice" Z2 of points with integer coordinates in the Cartesian
15 An ingenious innocent might observe that, each real number bp.ing cancelled by
its opposite, the sum in question is "obviously" 0 ...
II - Convergence: Discrete variables
1 = IOglO(10) = 2 54 log 10 (1 + h) rv 2 54 Mh.
To calculate loglO 2 for example, he again extracted 54 square roots of 2, and
found
1.000000000000000038477397965583 10 = 1 + h'
and since
finally he found
10glO 2 = h' /h = 0.30102 99956 63881 2.
This done, you start again with 3, 5, 7, etc. One may well suspect that, as
we shall see, more economical procedures were invented later.
We shall return to all this d propos the logarithmic functions.
Example 8. We have
(5.9)
limnl/n = 1.
Let us put nl/n = 1 + Xn where clearly Xn > O. The binomial formula
n = (1 + xn)n = 1 + n'Xn + ~n(n -1)x~ + ... ,
in which all the terms are> 0, shows that x; < 2/(n-1), an expression which
tends to O. For all r > 0, one thus has x; < r2 for n large, so 0 < Xn < r,
and thus Xn tends to 0, qed. (A sequence which tends to 0 is < r2 or r 624 for
n large since r2 > 0).
6 - The language of series
Though not basically different from that of sequences, the language of series is
frequently convenient, principally because many individual functions can be
represented more naturally by series than by sequences; their theory scarcely
appeared before Cauchy's time, the 1820s, while they are ubiquitous after
1650.
The fundamental problem is to give a meaning to a sum of an infinite
number of terms, imposing, of course, reasonable conditions; we do not propose to explain what the sum 1 -2 + 3 - 4 + 5 - ... might signify, let alone
the sum of all the real numbers 15 , for it is prudent to restrict oneself to sums
having no more than a countable infinity of terms. The terms of such a sum
may, by hypothesis, be put in the form of a sequence Ul, U2, .... Most often,
they are given in advance in this form, but there are also cases where it would
be quite artificial to order the terms of the sum as a sequence. Consider for
example the "lattice" Z2 of points with integer coordinates in the Cartesian
15 An ingenious innocent might observe that, each real number bp.ing cancelled by
its opposite, the sum in question is "obviously" 0 ...