§2. Uniform convergence
223
There is no problem for Ixl < 1 since we are dealing with a power series
whose radius of convergence is clearly 1. To examine what happens when x
tends to 1, let us work on [0,1]. The series is then alternating with decreasing
terms. Writing Ln(x) for the nth partial sum of the series, the general result
of Chap. II, nO 13 then tells us that
(8.8)
IL(x) - Ln(x)1 ::; x n +1 /(n + 1) ::; l/(n + 1) for ° : : ; x::; l.
In consequence, Ln(x) converges uniformly to L(x) on this interval, whence
continuity on the closed interval [0, 1] and so on ]-1, 1] as stated. If one knows
that the function 10g(1 + x) is continuous for x > -1 and is represented by
the series (7) for Ixl < 1 (strict inequality), one can then pass to the limit
when x -4 1 - 0, whence
(8.9)
1 - 1/2 + 1/3 - 1/4 + ... = log 2.
It goes without saying that this formula is no better suited to the numerical
calculation of log 2 than Leibniz' formula for 7r/4 which evoked Newton's
irony.
One should also remark that, though the series (7) converges uniformly
on [0, 1], we have not proved that it converges normally there, for this is
patently false: a series with positive terms which majorises the moduli of the
terms of (7) for any x E [0,1] must, for x = 1, majorise the harmonic series
and so cannot possibly converge.
Example 5. Consider the square wave series. Clearly it cannot converge uniformly on the interval [0,7r] since its sum sex) is discontinuous at x = 7r /2;
Further, if you consider the subset of the plane included between the graphs
of sex) +r and sex) -r, with for example r = 7r/12, you will obtain the union
of the three following sets:
(i)
(ii)
(iii)
0::; x < 7r/2,
x=7r/2,
7r/2 < x ::; 7r,
7r/6 < y < 7r/3,
Iyl < 7r/12,
-7r/3 < y < -7r/6.
How can a graph lying entirely in this strip pass from the strip (i) to the
strip (iii) across (ii) while remaining continuous? A simple sketch will let
one understand the problem, which gives rise to the "Gibbs phenomenon",
called overshoot by the electricians. While waiting to read the chapter of this
book consecrated to Fourier series, the reader may practice by tracing the
graphs of the first partial sums of the series; this exercise presupposes only
elementary knowledge of derivatives and of the trigonometric functions, and
exhibits the phenomenon very clearly17. Convergence of the series, which the
17 One can also, of course, trace these curves with the help of computer programmes,
but this is not the method to learn mathematics; mathematics is not a black box
in whose interior phenomena are produced that one does not try to understand.
223
There is no problem for Ixl < 1 since we are dealing with a power series
whose radius of convergence is clearly 1. To examine what happens when x
tends to 1, let us work on [0,1]. The series is then alternating with decreasing
terms. Writing Ln(x) for the nth partial sum of the series, the general result
of Chap. II, nO 13 then tells us that
(8.8)
IL(x) - Ln(x)1 ::; x n +1 /(n + 1) ::; l/(n + 1) for ° : : ; x::; l.
In consequence, Ln(x) converges uniformly to L(x) on this interval, whence
continuity on the closed interval [0, 1] and so on ]-1, 1] as stated. If one knows
that the function 10g(1 + x) is continuous for x > -1 and is represented by
the series (7) for Ixl < 1 (strict inequality), one can then pass to the limit
when x -4 1 - 0, whence
(8.9)
1 - 1/2 + 1/3 - 1/4 + ... = log 2.
It goes without saying that this formula is no better suited to the numerical
calculation of log 2 than Leibniz' formula for 7r/4 which evoked Newton's
irony.
One should also remark that, though the series (7) converges uniformly
on [0, 1], we have not proved that it converges normally there, for this is
patently false: a series with positive terms which majorises the moduli of the
terms of (7) for any x E [0,1] must, for x = 1, majorise the harmonic series
and so cannot possibly converge.
Example 5. Consider the square wave series. Clearly it cannot converge uniformly on the interval [0,7r] since its sum sex) is discontinuous at x = 7r /2;
Further, if you consider the subset of the plane included between the graphs
of sex) +r and sex) -r, with for example r = 7r/12, you will obtain the union
of the three following sets:
(i)
(ii)
(iii)
0::; x < 7r/2,
x=7r/2,
7r/2 < x ::; 7r,
7r/6 < y < 7r/3,
Iyl < 7r/12,
-7r/3 < y < -7r/6.
How can a graph lying entirely in this strip pass from the strip (i) to the
strip (iii) across (ii) while remaining continuous? A simple sketch will let
one understand the problem, which gives rise to the "Gibbs phenomenon",
called overshoot by the electricians. While waiting to read the chapter of this
book consecrated to Fourier series, the reader may practice by tracing the
graphs of the first partial sums of the series; this exercise presupposes only
elementary knowledge of derivatives and of the trigonometric functions, and
exhibits the phenomenon very clearly17. Convergence of the series, which the
17 One can also, of course, trace these curves with the help of computer programmes,
but this is not the method to learn mathematics; mathematics is not a black box
in whose interior phenomena are produced that one does not try to understand.