1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series (a) an { 0 lim - = n->oo n 00 1 . lnn (b) 1m --= 0 n~oo n ifO<a:::;l if a> 1 *20. If an > 0 and ...

186 Chapter4 From a more technical (but less intuitive) standpoint, a series is defined as an ordered pair of sequences, namely, ...

Sequences and Series 187 Proof Since Sm - Sn = an+1 + · · · +am ( 4. 7-1) follows at once from the Cauchy condition for sequence ...

188 Chapter4 In the convergence case it is easily seen that Theorem 4. 7 The series I;~ an converges iff both series of real ter ...

Sequences and Series 189 4.8 Some Properties of Series of Complex Terms Theorem 4.8 The sum of an absolutely convergent series d ...

190 Chapter4 This example also shows that the "dissociative" property is not valid for convergent series: i.e., replacement of e ...

Sequences and Series 191 By assumption, the series 00 00 and are convergent. The series of the products of their terms as in ( 4 ...

192 Chapter4 Theorem 4.14 The series ( 4.8-5) obtained by Cauchy's rule is convergent whenever one of the series A= I:~ an, B =I ...

Sequences and Series 193 and C = AB. In particular, if the factors are convergent, their prod- uct is summable ( C, 1), and C = ...

194 Chapter4 fails, unless Kn :::; 0 for n > N, in which case the series diverges by the first form of Kummer's test, provide ...

Sequences and Series taking Dn = nlnn in (c). Then Dn+1 = (n + 1) ln(n + 1) = (n + 1) (inn+ ln ( 1 + ~)] =(n+l) [inn+~ +O(n-^2 ) ...

196 Chapter4 so that I: lanl converges if f3 > 1 and diverges if f3 < 1. For the case (3 = 1, we have, using (f), lim lnn ...

Sequences and Series 197 so, by Gauss's test, the series converges absolutely if Re( c + 1 - a - n) > 1, or Re(c - a - b) > ...

198 Chapter 4 Example Consider the sequence defined on C by Clearly, since un(z)=(2+n~l)~, n = 0, 1,2, ... un(z)-+ 2z, z EC, asn ...

Sequences and Series limz->zo,zEB Un(z) = Ln. Then (a) limn->oo Ln = L (L finite) (b) limz->zo,zEB f(z) = L 199 The f ...

200 Chapter4 By taking in (4.10-2) a fixed value of n > max(N 1 ,N2), we obtain, using (4.10-3) to (4.10-5), lf(z)-Ll<E pr ...

Sequences and Series 201 At any point zo E D where the series of the absolute values of the terms, namely, 00 L lun(z)I = luo(z) ...

202 Chapter4 for n > N., arbitrary p, and all z E E. Then the sufficient part of the Cauchy condition yields the absolute and ...

Sequences and Series 203 Remark In general, a test for convergence becomes a test for uniform convergence in some set Di if it i ...

204 By elementary algebra we have 1 - zn+l Sn(z)=l+z+···+zn= --- 1-z provided that z -:/:- 1. Chapter4 1-z 1-z Obviously, the se ...