§3. Bolzano-Weierstrass and Cauchy's criterion
237
Corollary 2. Let (vn ) be a sequence of positive numbers decreasing to o .
. Then the series E vnzn converges on the set
(11.6)
X : Izl ~ 1, z #1
and, for any number r > 0, uniformly on the set
(il. 7)
X(r) : Izl ~ 1,11- zl ~ ri
its sum is continuous on X.
Since Izl ~ 1 and z #1 we have
11 + z + ... + zn-11 = 1(1 - zn)/(1 - z)1 ~ 2/11 - zl
for any n, whence convergence on X by Dirichlet's Theorem (make Un = zn).
Further, the inequality (2) shows, at the limit, that the remainder rp(z) of
the series, is, in modulus, majorised by 4vp /ll - Zli if one works on the set
X(r), equivalently, removes from the closed disc Izl ~ 1 a neighbourhood
of the point 1 where the situation may become catastrophic if the series Vn
diverges, one then has
on X(r) for all n > p, whence
Now, for r given, the right hand side is arbitrarily small for p large, qed.
-i
fig. 12.
Exercise: show that any compact K c X is contained in an X(r) and
deduce from this that the series converges uniformly on K.