Principles of Mathematics in Operations Research

13.1 Series 177

13.1.2 Operations on Series

Proposition 13.1.6 IfJ2T=o uk and SfcLo Vk are convergent series and a €

C, then the series Yl'kLo aUk and 12 <kLo(Uk -'- Vk) are a^so convergent and we

have

oo oo oo oo oo

^crafe = a^Mfc and Y^(uk±vk) = y^uk±y^Vk-

k=0 k=0 fc=0 fc=0 fc=0

Proof. Indeed,

J2T auk = linin^oc ^o auk = «lim„->oo J2ouk=a Eo° ufc> and

X)~(«fc ± vk) = linin^oo J]g (wfc ± Wfe) = limnr+00 ^)o ufc ± lim.n_>oo So wfc =

J2™uk±J2'o'vk- D

Remark 13.1.7 .ft should be stressed that, generally speaking, the conver-

gence of ^^° Uk ± X^o° Wfc does not imply the convergence of each of the series

££1:0 wfc and SfcLo^*" w'l*c'^1 can be confirmed by the example below:

(a-a) + (a-a)-\ , Va e C.

13.1.3 Tests for positive series

Theorem 13.1.8 (Comparison Tests) Let there be given two series

oo oo

(i) ^Uk and (ii) ^«fc

o o

with nonnegative terms.

(a) If Uk < Vk, VA;, the convergence of series (ii) implies the convergence of

series (i) and the divergence of series (i) implies the divergence of series

(ii).

(b) If linifc-Kx, ^ = A > 0, then series (i) and (ii) are simultaneously con-

vergent and divergent.

Proof. Exercise! D

Theorem 13.1.9 (D'Alembert's Test) Let there be a positive series

oo

Y^Uk 3 uk > 0, Vk = 0,1,...

o

(a) If ^^ < q < 1, VA;, then the series J2™ uk is convergent. If ^^ > 1,

then the series ]Po° uk *s divergent.

(b) If linifc-^oo ^^ = q then the series J^'o' Uk *s convergent for q < 1 and

divergent for q > 1.

Proof. We treat the cases individually.