082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

6.2 Sequences 205
Ageneralgeometricprogressionhasfirsttermaandcommonratiorandcantherefore
be writtenas
a,ar,ar 2 ,ar 3 ,...
anditiseasytoseethatthekthtermisar k−1 .Allgeometricprogressionscanbewritten
recursively asx[k] =rx[k −1].
Geometric progression:a,ar,ar 2 ,...
a = first term,r = common ratio,kth term =ar k−1
6.2.3 Moregeneralsequences
We have already metanumber of infinite sequences. For example,
(1) x[k]=2,4,6,8,...
(2) x[k] = 1, 1 2 , 1 4 ,...
In case (1) the terms of the sequence go on increasing without bound. We say the sequenceisunbounded.Ontheotherhand,incase(2)itisclearthatsuccessivetermsget
smallerandsmallerandask → ∞,x[k] → 0.Thenotionofgettingcloserandcloserto
afixedvalueisveryimportantinmathematicsandgivesrisetotheconceptofalimit.In
case(2)wesay‘thelimitofx[k]asktendstoinfinityis0’andwewritethisconciselyas
lim x[k] = 0
k→∞
We say that the sequence converges to 0, and because its terms do not increase without
bound wesay itisbounded.
More formally, we say that a sequencex[k] converges to a limitl if, by proceeding
far enough along the sequence, all subsequent terms can be made to lie as close tol as
wewish.Whenever a sequence isnotconvergent itissaidtobedivergent.
Itispossibletohavesequences whicharebounded butnevertheless donotconverge
toalimit. The sequence
x[k] = −1,1,−1,1,−1,1,...
clearly fails to have a limit ask → ∞ although it is bounded, that is its values all lie
within a given range. This particular sequence issaidtooscillate.
Itispossibletoevaluatethelimitofasequence,whensuchalimitexists,fromknowledgeofitsgeneralterm.Tobeabletodothiswecanmakeuseofcertainrules,theproofs
of which arebeyond the scope of thisbook, but which wenow state:
Ifx[k]andy[k]aretwosequencessuchthatlim k→∞
x[k] =l 1
,andlim k→∞
y[k] =l 2
,
wherel 1
andl 2
arefinite,then:
(1) Thesequencegiven byx[k] ±y[k] has limitl 1
±l 2
.
(2) Thesequencegiven bycx[k], wherecisaconstant, has limitcl 1
.
(3) Thesequencex[k]y[k] has limitl 1
l 2
.
(4) Thesequence x[k]
y[k] has limit l 1
providedl
l 2 ≠ 0.
2