4.1. SEQUENCES AND THE GEOMETRIC SERIES 125
Problem 4.12 Compute the limit of each of the following sequences or give a reason why the
limit does not exist. Assume n D 1; 2; : : :
1.
˚
x
n
D tan
1
.n/
2.
n
y
n
D sin
2
n
o
3.
z
n
D
n
2
n C1
4.
f
y
n
D sin
.
n
/
g
5.
z
n
D
3n
2
n
2
C 1
6.
z
n
D
cos.n/
n C1
Problem 4.13 Do the calculation to prove the infinite geometric series formula from the finite
one: see Knowledge Box 4.7.
Problem 4.14 Compute the following sums or give a reason they fail to exist.
1.
20
X
kD0
1:2
k
2.
1
X
nD0
1
3
n
3.
1
X
nD0
2
1
7
n
4.
1
X
nD0
1
4
n
5.
1
X
nD0
3
3
2
n
6.
1
X
nD0
0:05
n
7.
1
X
nD0
112
.
0:065
/
n
8.
1
X
nD0
2 .1/
n
Problem 4.15 Compute the following sums or give a reason they fail to exist.
1.
24
X
kD12
3
n
2.
15
X
kD5
2
n
3.
30
X
kD3
1:2
n
4.
20
X
kD10
4:5
n
5.
100
X
kD90
1:1
n
6.
22
X
kD1
7
n
Problem 4.16 A swinging pendulum is losing energy. Its first swing is 2 m long, and each
after that is 0.9985 times as long as the one before it. Estimate the total distance traveled by the
pendulum.
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