Principles of Mathematics in Operations Research

Solutions 285

Thus, it is convergent.

b) £r f^>

where a

>

0:

It is convergent for x = 0. Let us assume that 0 < x < 1.

*"+xa f k \a

= -—r— = x I 1 —> x as k —> oo.

«fc f£ \k + lj

Thus, it is convergent. If x > 1, it is divergent since ^^ —>• x as fc —> oo. If

x = 1, we have Ei° fc~a, a > 0. Then,

lim — , , Un+l

n->oo «„ \k + 1

The series is convergent when a > 1 and divergent when a < 1. In the special

case where a = 1, it is (f), the harmonic series which is divergent.

c) J2T(ei - 1):

e* - lsa^asA:—• oo. Thus, it is divergent (see part f) below).

d)£rm(l + £):

In (l + jr) ss jr as k —> oo. Thus, it is divergent (see part f) below).

e) Er qk+^, where q- > 0:

t/u^ = ^^1 +fc -> 5 as k -> oo. Thus, it is convergent for 0 < q < 1 and

divergent for q > 1. If q = 1, then w^ = 1 and E 1 is divergent.

>oo 1 ,

oEra

^=*±I = -*--> las *->«>.

"A: £ fc + 1

The Harmonic series is divergent!

13.3
a) For each object i = 1,..., n, either it is selected or not; that is Xi € Si =
{0,1}. Then,
n
5 (x) = JJ(

a:0

+

;cl

)

=

(

1 + x

)

n

'

Without loss of generality, we may assume that r — E xi objects are selected.
We know from Problem 1.3.a) that the number of distinct ways of selecting