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

6.2 Sequences 201
6.2 SEQUENCES
Asequenceisasetofnumbersorterms,notnecessarilydistinct,writtendownina
definite order.
For example,
1,3,5,7,9 and 1, 1 2 , 1 4 , 1 8 , 1
16
are both sequences. Sometimes we use the notation ‘...’ to indicate that the sequence
continues. For example, the sequence 1,2,3,...,20 is the sequence of integers from 1
to 20 inclusive. These sequences have a finite number of terms but we shall frequently
dealwithonesinvolvinganinfinitenumber ofterms.Toindicatethatasequencemight
go on for ever we can use the ... notation. Thus
and
2,4,6,8,...
1,−1,1,−1,...
can be assumed tocontinue indefinitely.
Ingeneral situations we shall writeasequence as
x[1],x[2],x[3],...
or more compactly,
x[k]
k=1,2,3,...
An alternative notation is
x 1
,x 2
,x 3
,...
Theformernotationisusuallyusedinsignalprocessingwherethetermsinthesequence
representthevalues ofthesignal.Thelatternotationarisesinthenumerical solutionof
equations. Hence both forms will be required. Often x[1] will be the first term of the
sequence although thisisnotalways the case.The sequence
...,x[−3],x[−2],x[−1],x[0],x[1],x[2],x[3],...
isusually writtenas
x[k]
k=...,−3,−2,−1,0,1,2,3,...
Acompletesequence,asopposedtoaspecifictermofasequence,isoftenwrittenusing
braces,for example
{x[k]} =x[1],x[2],...
although it is common to writex[k] for both the complete sequence and a general term
inthesequencewhenthereisnoconfusion,andthisistheconventionweshalladoptin
thisbook.
Asequencecanalsoberegardedasafunctionwhosedomainisasubsetofthesetof
integers.For example, the function defined by
x:N→R
x:k→ 3k
2