§4. Differentiable functions
265
17 - Sequences and series of differentiable functions
The classical analysts, notably Euler but also Fourier a good half century
later, seem to have believed that, if a sequence or series of differentiable
functions converges, one will, on differentiating term-by-term as for a finite
sum, again obtain a convergent sequence or series. Fourier's own square wave
series rebutted this conjecture; its sum is neither differentiable nor even continuous for x = 7r /2 and, moreover, differentiating it leads, as we have seen,
to bewildering results. Fourier, in reality, was trying to represent the periodic
function equal to 1 for Ixl < 7r/2 and to -1 for 7r/2 < Ixl < 7r as a series of
the form al cos x - a3 cos 3x + a5 cos 5x - ... , with coefficients to be determined. To do this he set x = 0 in the given series and in those obtained by
differentiating term-by-term ad libitum; he found the relations
al - a3 + a5 - ...
1
al - 32a3 + 52a5 - . . .
0
al - 34a3 + 54a5 - . . .
0
etc. So now he had to determine the coefficients from a system of an infinite
number of linear equations in an infinite number of unknowns. To solve this,
Fourier simplified it by replacing, for given n, the unknowns a2n+3, a2n+5,
etc. by 0; working then with only the first n equations he obtained a fully
orthodox system which he solved explicitly by computations within the reach
of a present day candidate to the Ecole polytechnique. He then made n tend
to infinity in his formulae, and thanks to the expansion of 7r as an infinite
product (Wallis' formula) finally found, up to the factor 4/7r, the square wave
series cosx - cos(3x}/3 + cos(5x}/5 - ....
But had he wanted to verify the result of his calculations he would have
had to substitute these values in his infinite system of linear equations and
would then have obtained the formulae
1 - 1/3 + 1/5 - 1/7 + .. .
1 - 3 2 /3 + 5 2 /5 - 7 2 /7 + .. .
1 - 3 4 /3 + 5 4 /5 - 7 4 /7 + .. .
7r/4
o
0;
the first one, Leibniz' formula, is correct, but the others are still more foolish
than the marvels of the harmonic series. The mathematicians of the XIX th
century have discovered even more bewildering counterexamples than this
last: a sum or series of indefinitely differentiable functions, even if uniformly
convergent, can have a sum not admitting a derivative anywhere. Bolzano
constructed one, though this was not discovered until 1930. Riemann considered the series Lsin(n 2 x}/n 2 and tried, without success, to show that its
sum has no derivative (in fact, in 1970 it was proved differentiable at certain
points, for example x = m7r /n with m and n odd, but not if x/7r is irrational).
265
17 - Sequences and series of differentiable functions
The classical analysts, notably Euler but also Fourier a good half century
later, seem to have believed that, if a sequence or series of differentiable
functions converges, one will, on differentiating term-by-term as for a finite
sum, again obtain a convergent sequence or series. Fourier's own square wave
series rebutted this conjecture; its sum is neither differentiable nor even continuous for x = 7r /2 and, moreover, differentiating it leads, as we have seen,
to bewildering results. Fourier, in reality, was trying to represent the periodic
function equal to 1 for Ixl < 7r/2 and to -1 for 7r/2 < Ixl < 7r as a series of
the form al cos x - a3 cos 3x + a5 cos 5x - ... , with coefficients to be determined. To do this he set x = 0 in the given series and in those obtained by
differentiating term-by-term ad libitum; he found the relations
al - a3 + a5 - ...
1
al - 32a3 + 52a5 - . . .
0
al - 34a3 + 54a5 - . . .
0
etc. So now he had to determine the coefficients from a system of an infinite
number of linear equations in an infinite number of unknowns. To solve this,
Fourier simplified it by replacing, for given n, the unknowns a2n+3, a2n+5,
etc. by 0; working then with only the first n equations he obtained a fully
orthodox system which he solved explicitly by computations within the reach
of a present day candidate to the Ecole polytechnique. He then made n tend
to infinity in his formulae, and thanks to the expansion of 7r as an infinite
product (Wallis' formula) finally found, up to the factor 4/7r, the square wave
series cosx - cos(3x}/3 + cos(5x}/5 - ....
But had he wanted to verify the result of his calculations he would have
had to substitute these values in his infinite system of linear equations and
would then have obtained the formulae
1 - 1/3 + 1/5 - 1/7 + .. .
1 - 3 2 /3 + 5 2 /5 - 7 2 /7 + .. .
1 - 3 4 /3 + 5 4 /5 - 7 4 /7 + .. .
7r/4
o
0;
the first one, Leibniz' formula, is correct, but the others are still more foolish
than the marvels of the harmonic series. The mathematicians of the XIX th
century have discovered even more bewildering counterexamples than this
last: a sum or series of indefinitely differentiable functions, even if uniformly
convergent, can have a sum not admitting a derivative anywhere. Bolzano
constructed one, though this was not discovered until 1930. Riemann considered the series Lsin(n 2 x}/n 2 and tried, without success, to show that its
sum has no derivative (in fact, in 1970 it was proved differentiable at certain
points, for example x = m7r /n with m and n odd, but not if x/7r is irrational).