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

206 Chapter 6 Sequences and series
Furthermore, wecan always assume that
lim
k→∞
1
= 0 forany constantm > 0
km Example6.8 Find,ifpossible, the limitofeachofthe following sequences,x[k].
(a) x[k] = 1 k
(b) x[k]=5
(c) x[k] = 3 + 1 k
(d) x[k] = 1
k +1
(e) x[k]=k 2
k=1,2,3,4,...
k=1,2,3,4,...
k=1,2,3,4,...
k=1,2,3,4,...
k=1,2,3,4,...
Solution (a) The sequencex[k] isgiven by
1, 1 2 , 1 3 , 1 4 ,...
Successivetermsgetsmallerandsmaller,andask → ∞,x[k] → 0.Byproceeding
far enough along thesequence wecan get asclose tothelimit0as wewish.Hence
1
lim x[k] = lim
k→∞ k→∞ k = 0
(b) The sequencex[k] isgiven by 5,5,5,5,.... This sequence has limit5.
(c) Thesequence3,3,3,3,...haslimit3.Thesequence1, 1 2 , 1 3 ,...haslimit0.Therefore,usingrule(1)
wehave
lim
k→∞ 3 + 1 k =3+0=3
The terms of the sequence x[k] = 3 + 1 k are given by 4,31 2 ,31 3 ,...,andby
proceeding far enough along we can make all subsequent terms lie as close to the
limit3as wewish.
(d) The sequencex[k] = 1
k +1 ,k=1,2,3,4,...,isgivenby
1
2 , 1 3 , 1 4 ,...
and has limit0.
(e) The sequence x[k] = k 2 ,k = 1,2,3,4,..., is given by 1,4,9,16,..., and
increases withoutbound. This sequence has no limit-- itisdivergent.
Example6.9 Given a sequencewith generaltermx[k] = k −1
k +1 , find lim k→∞ x[k].
Solution It is meaningless simply to writek = ∞ to obtain lim k→∞
x[k] = ∞ −1 , since such
∞ +1
a quantity is undefined. What we should do is try to rewritex[k] in a form in which we