224
III - Convergence: Continuous variables
7tJ4+ r 3_.
7tJ4
7tJ4-r
fig. 10.
simplistic criteria of Chap. II are inadequate to establish, will be established
in nO 11.
Example 6. Consider the Weierstrass series
(Chap. II, nO 23) for k ~ 3. Suppose that z remains in a disc D : Izl :::; R,
of finite radius and let us omit from the series the terms, finite in number,
for which Iwl < 2R. Then clearly Iz - wi ~ Iwl/2 for all zED and all w. In
consequence, the series is, on D, dominated up to a factor 2k by the series
L 1/lwl k which, as we have seen, converges. The series fk is thus normally
convergent on D (after subtracting the terms for which Iwl < 2R).