§ 1. Convergent sequences and series
71
The sequence
Un = 1 + 1/2 + 1/3 + ... + l/n,
is divergent or, what comes to the same for an increasing sequence, its terms
increase above any bound as we shall see in n° 7. Even on choosing an index n
as hyperastronomic as 10 100 , one finds a value only of the order of 230, a result
which, numerically, is perfectly compatible with the false hypothesis that the
sequence converges to 23l.
Mathematicians such as Newton, Stirling or Euler who worked to obtain
these numerical estimates had methods more ... intellectual than that of
churning for an undetermined time (the experts will estimate it for us) the
"acres of computers" of the American National Security Agency, at the risk
of compromising the said security for the idle amusement of mathematicians.
They already have trouble in finding two prime numbers p and q when the
product pq, all one tells them, has a hundred or so digits.
To conclude these generalities, let us observe that the definition of a limit
supposes that the limit to be obtained is known. It is nevertheless possible
to decide on the convergence of a sequence without knowing the limit in
advance. In this direction one notes that if one has d( u, Un) < r /2 for all
n > N, one will also have, by the triangle inequality,
d(up,uq) < r once p > Nand q > N.
We shall show in the following chapter that this necessary condition for convergence is also sufficient; this is Cauchy'8 general criterion of convergence,
known before him to Bolzano, and which neither really proved. One can make
the result seem very plausible by choosing numbers r of the form lO-n; if
decimal numeration did not exhibit the bizarre behaviour to which we have
already alluded, the preceding inequality would show that starting from a
certain rank, the first n digits of the terms of the sequence would no longer
change, and this would demonstrate convergence.
Let us now give some examples of convergent sequences; the first are
almost trivial, but those following will be very useful in the sequel.
Example 1. The "constant" sequence u, u, u, ... converges to u.
Example 2. One has
(5.2)
lim1/n = 0,
since the relation 11/nl < r can be written as nr > 1 so is satisfied for large n
by Archimedes' axiom.
Example 3. One has
lim_n_ = 1
n+1
since 11 - n/{n + 1)1 = l/{n + 1) tends to ° by the preceding example.
71
The sequence
Un = 1 + 1/2 + 1/3 + ... + l/n,
is divergent or, what comes to the same for an increasing sequence, its terms
increase above any bound as we shall see in n° 7. Even on choosing an index n
as hyperastronomic as 10 100 , one finds a value only of the order of 230, a result
which, numerically, is perfectly compatible with the false hypothesis that the
sequence converges to 23l.
Mathematicians such as Newton, Stirling or Euler who worked to obtain
these numerical estimates had methods more ... intellectual than that of
churning for an undetermined time (the experts will estimate it for us) the
"acres of computers" of the American National Security Agency, at the risk
of compromising the said security for the idle amusement of mathematicians.
They already have trouble in finding two prime numbers p and q when the
product pq, all one tells them, has a hundred or so digits.
To conclude these generalities, let us observe that the definition of a limit
supposes that the limit to be obtained is known. It is nevertheless possible
to decide on the convergence of a sequence without knowing the limit in
advance. In this direction one notes that if one has d( u, Un) < r /2 for all
n > N, one will also have, by the triangle inequality,
d(up,uq) < r once p > Nand q > N.
We shall show in the following chapter that this necessary condition for convergence is also sufficient; this is Cauchy'8 general criterion of convergence,
known before him to Bolzano, and which neither really proved. One can make
the result seem very plausible by choosing numbers r of the form lO-n; if
decimal numeration did not exhibit the bizarre behaviour to which we have
already alluded, the preceding inequality would show that starting from a
certain rank, the first n digits of the terms of the sequence would no longer
change, and this would demonstrate convergence.
Let us now give some examples of convergent sequences; the first are
almost trivial, but those following will be very useful in the sequel.
Example 1. The "constant" sequence u, u, u, ... converges to u.
Example 2. One has
(5.2)
lim1/n = 0,
since the relation 11/nl < r can be written as nr > 1 so is satisfied for large n
by Archimedes' axiom.
Example 3. One has
lim_n_ = 1
n+1
since 11 - n/{n + 1)1 = l/{n + 1) tends to ° by the preceding example.