212
III - Convergence: Continuous variables
Consider the increases which these three functions receive when one
increases x by an infinitely small quantity a [for Cauchy, this means
that a tends to 0]. The increase in Sn will be, for all possible values
of n, an infinitely small quantity; and that in rn will become imperceptible at the same time as r n, if one attributes a very considerable
value to n. In consequence, the increase in the function s must be an
infinitely small quantity.
Here we have an excellent example of the errors into which the use of
vague language can lead a first class mathematician, for Cauchy did not
write precise inequalities, at least in this passage. The first ambiguity occurs
at the point where he claims that "the increase in Sn will be, for all possible
values of n, an infinitely small quantity". If Cauchy wants to say by this
that for each n, sn(x + a) - sn(x) tends to 0 with a he is perfectly right,
since the Sn, being finite sums of continuous functions, are continuous. If,
however, he wants to say that for lal < 8 one has ISn(x + a) - sn(x)1 < c:
for all n simultaneously, in other words that the family Sn is equicontinuous,
the argument is false even if the limit function is continuous, as we have seen
above.
The second error is to claim that the increase in r n "will become imperceptible at the same time as rn if one attributes a very considerable value to
nj" since, for x given, Irn(x)1 < c: for n > N(x) - this is simple convergence-,
Cauchy's argument amounts to saying that for lal < 8, one has Irn(x+a)1 < c:
for all n > N(x), which is local uniform convergence as mentioned at the beginning of the preceding nO j this would follow from continuity of the limit
function, a property that one may not use to establish continuity of itself ...
It was only in 1853 that Cauchy corrected his ''theorem'' of 1821 by exploiting uniform convergence quite correctly, though without naming this
concept. But his error was detected rapidly. The young Norwegian mathematician Abel, who was living in Paris, provided, in a letter of 1825 to a
friend (published in 1839 in his Complete Works), as Dugac tells us, a counterexample in the series
(6.1)
.
. 2 /2
. 3 /3
_ {X/2 if
sm x - sm x + sm x - ... - 0
if
Ixl < 'Tr,
Ixl = 'Trj
one of those exhibited by Fourier at the beginning of his Theorie analytique
de la chaleur, after the square wave series which Abel would have also been
well able to use, and which Cauchy, at Paris where Fourier still lived, should
have known, or maybe had forgotten. One also sees Abel, in the same letter,
complain about divergent series and unjustified manipulations of series of
functions, such as, for example, differentiating term-by-term as if dealing with
finite sums (see nO 17)j Abel observes that if one differentiates the formula
(6.2)
x/2 = sinx - sin2x/2 + sin3x/3 - ... ,
one obtains the relation
III - Convergence: Continuous variables
Consider the increases which these three functions receive when one
increases x by an infinitely small quantity a [for Cauchy, this means
that a tends to 0]. The increase in Sn will be, for all possible values
of n, an infinitely small quantity; and that in rn will become imperceptible at the same time as r n, if one attributes a very considerable
value to n. In consequence, the increase in the function s must be an
infinitely small quantity.
Here we have an excellent example of the errors into which the use of
vague language can lead a first class mathematician, for Cauchy did not
write precise inequalities, at least in this passage. The first ambiguity occurs
at the point where he claims that "the increase in Sn will be, for all possible
values of n, an infinitely small quantity". If Cauchy wants to say by this
that for each n, sn(x + a) - sn(x) tends to 0 with a he is perfectly right,
since the Sn, being finite sums of continuous functions, are continuous. If,
however, he wants to say that for lal < 8 one has ISn(x + a) - sn(x)1 < c:
for all n simultaneously, in other words that the family Sn is equicontinuous,
the argument is false even if the limit function is continuous, as we have seen
above.
The second error is to claim that the increase in r n "will become imperceptible at the same time as rn if one attributes a very considerable value to
nj" since, for x given, Irn(x)1 < c: for n > N(x) - this is simple convergence-,
Cauchy's argument amounts to saying that for lal < 8, one has Irn(x+a)1 < c:
for all n > N(x), which is local uniform convergence as mentioned at the beginning of the preceding nO j this would follow from continuity of the limit
function, a property that one may not use to establish continuity of itself ...
It was only in 1853 that Cauchy corrected his ''theorem'' of 1821 by exploiting uniform convergence quite correctly, though without naming this
concept. But his error was detected rapidly. The young Norwegian mathematician Abel, who was living in Paris, provided, in a letter of 1825 to a
friend (published in 1839 in his Complete Works), as Dugac tells us, a counterexample in the series
(6.1)
.
. 2 /2
. 3 /3
_ {X/2 if
sm x - sm x + sm x - ... - 0
if
Ixl < 'Tr,
Ixl = 'Trj
one of those exhibited by Fourier at the beginning of his Theorie analytique
de la chaleur, after the square wave series which Abel would have also been
well able to use, and which Cauchy, at Paris where Fourier still lived, should
have known, or maybe had forgotten. One also sees Abel, in the same letter,
complain about divergent series and unjustified manipulations of series of
functions, such as, for example, differentiating term-by-term as if dealing with
finite sums (see nO 17)j Abel observes that if one differentiates the formula
(6.2)
x/2 = sinx - sin2x/2 + sin3x/3 - ... ,
one obtains the relation