70
II - Convergence: Discrete variables
denoting by up its truncated or default decimal expansion of rank p; this is a
number of the form k/lO P , where k is an integer such that
up < U ::; up + lO-P;
with this definition, the default decimal expansion of the number 1 is
0.9999&c. Then
d( U, un) ::; lO-p for all n 2: p,
from which U is the limit of the Un. This example makes clear the fact that
we cannot define the real numbers without using the concept of limit in one
way or another. It also shows that if one tries to limit the domain of the
analysis to Q there will be multitudes of sequences which will not converge
because of the fact that their limit is irrational.
We mentioned above that when a sequence (un) of real numbers converges
to a limit u, the decimal expansion of the number Un has a strong tendency
to stabilise as n increases indefinitely. One might deduce that a convenient
experimental method to ascertain convergence, or to calculate the limit, of a
sequence is to examine numerically a sufficiently large number of terms. This
can spring several surprises on modern programmed calculators.
If one considers for example the sequence with general term
which, we shall see later, converges to the number
e = 2.71828 18284 590 ... ,
the base of the natural logarithms, one finds that
U4 = 2.44141 ... , U64 = 2.69734 ... , U1024 = 2.71696 ... ,
which indicates that Un approaches its limit value only very slowly, and that
one would have to choose enormous values of n to obtain even ten or so
decimal places of the number e; very luckily, the sequence with general term
1 + 1/2! + 1/3! + ... + l/n!
also converges to e, but with prodigious rapidity since the term l/(n + I)!
which one adds to Un to obtain U n +1 becomes microscopic very quickly.
Another example of a slowly convergent sequence:
Un
1/1.2 + 1/3.4 + 1/5.6 + ... + 1/(2n - 1)2n
1 - 1/2 + 1/3 - 1/4 + ... + 1/(2n - 1) - 1/2n.
It was already known by the end of the xvnth century that to calculate the
limit. namely log 2, exactly to 9 decimal places, quite a modest precision, one
must choose n > 10 8 ; see nO 13 for alternating series.
II - Convergence: Discrete variables
denoting by up its truncated or default decimal expansion of rank p; this is a
number of the form k/lO P , where k is an integer such that
up < U ::; up + lO-P;
with this definition, the default decimal expansion of the number 1 is
0.9999&c. Then
d( U, un) ::; lO-p for all n 2: p,
from which U is the limit of the Un. This example makes clear the fact that
we cannot define the real numbers without using the concept of limit in one
way or another. It also shows that if one tries to limit the domain of the
analysis to Q there will be multitudes of sequences which will not converge
because of the fact that their limit is irrational.
We mentioned above that when a sequence (un) of real numbers converges
to a limit u, the decimal expansion of the number Un has a strong tendency
to stabilise as n increases indefinitely. One might deduce that a convenient
experimental method to ascertain convergence, or to calculate the limit, of a
sequence is to examine numerically a sufficiently large number of terms. This
can spring several surprises on modern programmed calculators.
If one considers for example the sequence with general term
which, we shall see later, converges to the number
e = 2.71828 18284 590 ... ,
the base of the natural logarithms, one finds that
U4 = 2.44141 ... , U64 = 2.69734 ... , U1024 = 2.71696 ... ,
which indicates that Un approaches its limit value only very slowly, and that
one would have to choose enormous values of n to obtain even ten or so
decimal places of the number e; very luckily, the sequence with general term
1 + 1/2! + 1/3! + ... + l/n!
also converges to e, but with prodigious rapidity since the term l/(n + I)!
which one adds to Un to obtain U n +1 becomes microscopic very quickly.
Another example of a slowly convergent sequence:
Un
1/1.2 + 1/3.4 + 1/5.6 + ... + 1/(2n - 1)2n
1 - 1/2 + 1/3 - 1/4 + ... + 1/(2n - 1) - 1/2n.
It was already known by the end of the xvnth century that to calculate the
limit. namely log 2, exactly to 9 decimal places, quite a modest precision, one
must choose n > 10 8 ; see nO 13 for alternating series.