练习册以及答案.docx
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Exercises 9-1 Infinite Sequences
In problems1-6, determine whether the sequence
{ }
n
a
converges or diverges, find
lim
n
n
a
® ¥
.
1.
2
2
2 1
3 1
n
n
a
n n
-
=
+ -
Solution:
2
2
2
2
1
2
2 1
lim lim lim 2
3 1
3 1
1
n
n n n
n
n
a
n n
n n
® ¥ ® ¥ ® ¥
-
-
= = =
+ -
+ -
, so
the sequence
{ }
1
n
n
a
¥
=
converges.
2.
1
sin
n
a
n
æ ö
=
ç ÷
è ø
Solution:
1
lim lim sin lim sin 0 0
n
n n n
a
n
® ¥ ® ¥ ® ¥
æ ö
= = =
ç ÷
è ø
, so
the sequence
{ }
1
n
n
a
¥
=
converges.
3.
( )
1
2
n
n
n
a
n
= -
+
Solution: Since
2
2
lim lim 1
2 2
k
k k
k
a
k
® ¥ ® ¥
= =
+
and
2 1
2 1
lim lim 1
2 3
k
k k
k
a
k
+
® ¥ ® ¥
+
= - = -
+
, we conclude that
the sequence
{ }
1
n
n
a
¥
=
diverges.
4.
sin(n )
n
a
p
=
Solution:
( )
lim lim sin lim 0 0
n
n n n
a n
p
® ¥ ® ¥ ® ¥
= = =
, so
the sequence
{ }
1
n
n
a
¥
=
converges.
5.
( )
2
7
5
n
n
n
a
-
=
Solution: Since
( 7) 7
lim lim lim 0
(25) 25
n
n
n
n
n n n
a
® ¥ ® ¥ ® ¥
- -
æ ö
= = =
ç ÷
è ø
, the
sequence
{ }
1
n
n
a
¥
=
converges.
6.
ln
n
n
a
n
=
Solution:
1
ln ln 2
lim lim lim lim lim 0
1
2
n
n n x x x
n x
x
a
n x x
x
¥
¥
® ¥ ® ¥ ® ¥ ® ¥ ® ¥
= = = = =
, so the
sequence
{ }
1
n
n
a
¥
=
converges.
7. Write the first four terms of the sequence
1 1
1 2
2,
2
n n
n
a a a
a
+
æ ö
= = +
ç ÷
è ø
.
Show that the sequence converges.
Solution:
1 2 3
4
1 2 3 1 3 4 17
2, 2 , ,
2 2 2 2 2 3 12
1 17 24 577
.
2 12 17 408
a a a
a
æ ö æ ö
= = + = = + =
ç ÷ ç ÷
è ø è ø
æ ö
= + =
ç ÷
è ø
1 1
1 2 2
2 2, 2.
2
n n n
n n
a a a a
a a
+
æ ö
= > = + ³ × =
ç ÷
è ø
Thus
2
n
a ³
for all n, meaning that
{ }
1
n
n
a
¥
=
has a lower
bound.
Therefore,
2
1
2
1 2 1
0
2 2
n
n n n n
n n
a
a a a a
a a
+
æ ö
-
- = - + = ³
ç ÷
è ø
,
so
{ }
1
n
n
a
¥
=
is decreasing.
By the Monotonic Sequence Theorem, this means that the
sequence
{ }
1
n
n
a
¥
=
converges.
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