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

222 Chapter 6 Sequences and series
TechnicalComputingExercises6.6
Write acomputer program to implement the simple
iteration method. Bycomparing successive estimates
the program should check whether convergence is
taking place.
REVIEWEXERCISES6
1 Write down andgraph the firstfive terms ofthe
sequencesx[k] defined by
(a) x[k] = (−1) k , k =0,1,2,3,...
(b) x[k] = (−1)k
2k+1 , k=0,1,2,3,...
2 Find expressions forthekth termsofthe sequences
whose firstfiveterms are
(a) 1, 9, 17, 25,33
(b) −1, 1, −1, 1, −1
3 Forthe Fibonacci sequence
showthat
1,1,2,3,5,8,...
lim
k→∞
x[k +1]
x[k]
= 1 2
(
1 + √ )
5
[Hint:writex[k +1] =x[k] +x[k −1],form
x[k +1]/x[k] andtake limits.]
4 Usethe binomial theoremto expand (1 +x+x 2 ) 5 as
far asthe term inx 3 .
1
5 Usethe binomial theoremto expand
(3 +x) 3
in ascending powers ofxasfar asthe term inx 3 .
6 Thepower series expansion forln(1 +x) isgiven by
ln(1+x)=x− x2
2 + x3
3 − x4
4 +···
and isvalid for −1 <x 1. Take anumber ofvalues
ofxin thisinterval and obtain an approximate value
ofln(1 +x)by meansofthisseries.Compare your
answerswiththevaluesobtainedfromyourcalculator.
7 Bymultiplyingboth numerator anddenominatorof
√ √
k+1− k
by √ k+1+ √ k find
2
√ √
k+1− k
lim
k→∞ 2
8 Find lim
k→∞
(
) 3
3k−1
.
2k+7
2k 5 −3k 2
9 Find lim
k→∞ 7k 7 +2k .
10 Writedown the firsteight termsofthe series ∑ n
k=1 k.
Bynotingthat thisis an arithmeticseries showthat
n∑
k =
k=1
n(n +1)
2
11 Writedown the firstsixterms ofthe sequence defined
bythe recurrence relation
x[n +3] =x[n +2] −2x[n]
x[0] = 0 x[1] = 2 x[2] = 3
12 Findthe limit, ifitexists, ask → ∞ ofthe geometric
progression
when
a,ar,ar 2 ,...,ar k−1 ,...
(a) −1<r<1
(b) r>1
(c) r<−1
(d) r=1
(e) r=−1
13 An arithmeticseries has afirst termof4andthe 10th
termis 0.
(a) FindS 20 .
(b) IfS n = 0,findn.
14 Ageometricseries has
S 3 = 37
8
S 6 = 3367
512
Findthe firsttermand the common ratio.