Principles of Mathematics in Operations Research

284 Solutions

Problems of Chapter 13

13.1

Let there be given two series

A = 2~] Uk and B — \ vk

with nonnegative terms.

(a) // Wfc < Vk, yk, the convergence of series B implies the convergence of

series A and the divergence of series A implies the divergence of series B.

Suppose that B is convergent. Let S = Yl™ vk be finite.

n n

^2 uk < ]P

v

k < S, n = 0,1,...

o o

thus partial sum of A is bounded, hence it is convergent.

Suppose that A is divergent. Thus its nth partial sum increases indefinitely

together with n.

n n

0 0

Thus, nth partial sum of B increases indefinitely together with n, too.

That is, B is divergent.

(b) // lrnifc-^oo ^ = a > 0, then series A and B are simultaneously convergent

and divergent.

limfc-»oo % = a > 0, vk > 0, V*. Then,

Ve > 0 3N 9 a - e < — < a + e, Vfc > N.

Vk

=>• Vk{a — e) < Uk < Vk(a + e). If B is convergent, so is ^^° Vk(a — e).

Thus, A is convergent by (a). If B is divergent, so is ^^° vk{a — e). Thus,

A is divergent by (a).

13.2

a) Eo°°^0 ~k- X

It is convergent for x = 0. Let us assume that x > 0.

I*±! = J*+i)i =* _• 0 as A-• oo.

uk * k + 1