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

Review exercises 6 223
15 (a) Write down the seriesgiven by ∑ 5
r=1 r 2 .
(b) The sumofthe squaresofthe firstnwhole
numbers can be foundfrom the formula
n∑
r 2 n(n +1)(2n +1)
=
6
r=1
Use thisformula to find
(i) ∑ 5
r=1 r 2 , (ii) ∑ 10
r=1 r 2 , (iii) ∑ 10
r=6 r 2 .
16 (a) Writedown the series given by ∑ 6
r=1 r 3 .
(b) The sumofthe cubesofthe firstnwhole
numbers canbe foundfrom the formula
(
n∑
r 3 =
n(n +1)
2
r=1
Use thisformula to find
) 2
(i) ∑ 6
r=1 r 3 , (ii) ∑ 12
r=1 r 3 , (iii) ∑ 12
r=7 r 3 .
17 Thethirdterm ofan arithmeticprogression is 18.The
fifthterm is28. Find the sum of20terms.
18 (a) Find an expression forthe general term in the
sequence
2,5,10,17,...
(b) Define the sequence in terms ofarecurrence
relation.
19 (a) Show thatthe equationx 3 +2x −14 = 0 canbe
rearrangedintothe formx = 3√ 14 −2x.With
x 0 = 2 use simple iteration to findarootofthe
equation.
(b) Rearrange the equation 0.8sinx −0.5x = 0 into
the formx =g(x).Withx 0 = 2 use simple
iteration to findarootofthe equation.
(c) Rearrange the equationx 3 = 2e −x intothe form
x =g(x).Withx 0 = 0usesimpleiterationtofind
a rootofthe equation.
20 Writeoutexplicitlythe firstfour terms ofthe series
∞∑
m=0
(−1) m x 2m
2 2m (m!) 2 .
21 Writeoutexplicitlythe firstfour terms ofthe series
∞∑
m=0
(−1) m x 2m+1
2 2m+1 m!(m +1)! .
Solutions
1 (a) 1,−1,1, −1,1 (b) 1, − 1 3 , 1 5 , −1 7 , 1 9
2 (a) 8k−7 k 1 (b) (−1) k k 1
4 1+5x+15x 2 +30x 3
(
+···)
1
5 1 −x+ 2x2
27 3 − 10x3
27
7 0
8
9 0
27
8
11 0, 2, 3, 3, −1, −7
12 (a) 0 (b) nolimit (c) nolimit
(d) a (e) nolimit
13 (a) − 40
9
(b) 19
14 2, 3 4
15 (a) 1+4+9+16+25
(b) (i) 55 (ii) 385 (iii) 330
16 (a) 1+8+27+64+125+216
(b) (i) 441 (ii) 6084 (iii) 5643
17 1110
18 (a) x[k]=k 2 +1,k=1,2,3,...
(b)x[k+1]=x[k]+2k+1
19 (a) 2.13
(b) x = 1.6sinx,1.6
(c)x= 3 √
2e −x ,0.93
20 1− 1 4 x2 + 1 64 x4 − 1
2304 x6 +···
21 1 2 x − 1 16 x3 + 1
384 x5 − 1
18432 x7 +···