1550251515-Classical_Complex_Analysis__Gonzalez_

176 Chapter4

4. If Zn --7 Land {z~} is a rearrangement of {zn}, then also z~ --7 L.

5. If the sequence {Zn} is decomposed into two sequences { z~} and { z~}

(i.e., if either Zn = z;,. for some m, or Zn = z; for some p ), and

z~ --7 L, z~ --7 L, then also Zn --7 L.

6. If Zn = c (a constant) for n > N, then Zn --7 c.

7. If Zn --7 0 and { un} is a bounded sequence, then Zn Un --7 0.

8. If Zn --7 Land Wn --7 L', then C1Zn + c2Wn --7 c1L + c2L', ZnWn --7
   LL' and (zn/wn)--7 L/L', where c 1 ,c 2 are arbitrary constants, and
   Wn f:. 0, L' f:. 0 for the last property.

9. If Zn --7 L, then Zn --7 L.

10. If Zn --7 L, then Re Zn --7 Re L and Im Zn --7 Im L. Conversely, those

two statements imply that Zn --7 L.

11. If Zn --7 L, then lznl --7 ILi.

12. If Zn --7 L, then Arg Zn --7 Arg L, provided that L + ILi f:. 0.

13. Let f: D --7 <C and let a be an accumulation point of D. Then f has

a limit at a iff for each sequence {zn} converging to a, with Zn E D

and Zn f:. a for all n, the sequence {f(zn)} converges.

14. Let f: D --7 <C with a ED and a an accumulation point of D. Then

f is continuous at a iff for every sequence {zn} converging to a, with

Zn ED for all n, the sequence {f(zn)} converges to f(a).

15. If Zn --7 L and if {pn} is a sequence of complex numbers such that

<7 m = IP1 I + IP2 I + · · · + IPn I --7 oo, and such that

IP1 I + IP2 I + · · · + IPn I ~ k IP1 + P2 + · · · + Pn I

for a fixed k > 'o and every n, then

Wn= P1Z1 + P2Z2 + · · · + PnZn ~ ~ L

Pl+ P2 + · · · + Pn

( 4.3-1)

The sequence { Wn} is called the sequence of the weighted means of

the sequence {zn}.

In particular, if the Pn are positive real numbers such that p 1 +

· · ·+Pn --7 oo, the property holds [condition ( 4.3-1) now being trivially

satisfied]. As a particular case we have, taking Pn = 1 for all n,

Wn = Z1 + z2 + · · · + Zn --7 L

n

i.e., the sequence if the arithmetic means of the first n terms of {zn}

converges to the same limit as {zn}·

16. Suppose that {Pn} satisfies the same conditions as in property 15, and
    let Pn = P1 + P2 + · · · + Pn, Sn = Z1 + Z2 + · · · + Zn. If
    lim Sn - Sn-1 = L
    n-+oo Pn - Pn-1