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III - Convergence: Continuous variables
of derivatives L u~(x) is dominated, as above, by a series L Vi. One needs to
pass to the sums
SF(X) = L Ui(X), S(X) = L Ui(X).
iEF
iEI
Choose an a E K. The mean value theorem shows that
where m(K) is the length of K. Since L Ilu~IIK < +00, the series
L [Ui (x) - Ui (a) J converges normally on K for all a and x. If the series L Ui (a)
converges unconditionally then the series Ui(X) converges normally on K.
To show that one can differentiate it like a finite sum, consider the functions
qi(X) = [Ui(X) - ui(a)J/(x - a),
as in the proof of Theorem 19. They are defined on X = K - {a} and tend
to a limit u~(a) when x tends to a. Corollary 3 of the mean value theorem
shows on the other hand that
so that the series L qi(X) = [s(x) - s(a)J/(x - a) converges normally on X.
We can therefore pass to the limit when x tends to a as we saw above, which
shows that the sum s(x) is differentiable at a and that s'(a) = L u~(a), qed.
Consider for example, instead of the Riemann series of the preceding
nO, the double series /(s) = L(m 2 + n 2 )-s/2 with s real> 2 to ensure
convergence (Chap. II, n° 12). If we accept the formula (aX)' = aX log a of
Chap. IV, the derived series is
We leave it to the reader to show that it converges normally for s ~ a > 2 as
do all the successive derived series. The problem is only to show unconditional
convergence, because, the general term being a decreasing function of s, the
series g(a) dominates the series g(s) for all s ~ a. You may be guided by the
arguments of Chap. II, nO 12.