Math 310, HW 3, Due October 7, 2008, Solution
Instructor: Swatee Naik
1.
3.2: 10
Determine the following limits.
(a) lim
(
(3
√
n
)
(1
/
2
n
)
)
.
Note that (3
√
n
)
(1
/
2
n
)
= (
√
3)
(1
/n
)
n
(1
/
4
n
)
.
As shown in Example 3.1.11, (c), lim
(
(
√
3)
(1
/n
)
)
= 1
.
Since 1
≤
n
(1
/
4
n
)
≤
n
(1
/n
)
,
for every
n
∈
N
,
and by Example 3.1.11, (d), lim
(
n
(1
/n
)
)
=
1
,
by the Squeeze theorem, we have lim
(
n
(1
/
4
n
)
)
= 1
.
Theorem 3.2.3, (a) states that the product of two convergent sequences is convergent
and the limit of the product is product of the limits. Therefore, lim
(
(3
√
n
)
(1
/
2
n
)
)
= 1
.
(b) lim
(
(
n
+ 1)
1
/
(ln(
n
+1)
)
Note that since
n
+ 1 =
e
ln(
n
+1)
,
we have
x
n
= (
n
+ 1)
1
/
(ln(
n
+1)
=
e
for all
n
∈
N
.
So
this is the consant sequence (
e
) which converges to
e
.

x
n

e

= 0
< ± ,
for all
n
∈
N
and any given
±
>
0
.
2.
3.3: 2
Let
x
1
>
2 and
x
n
+1
:= 2

1
/x
n
for
n
∈
N
.
Show that (
x
n
) is bounded and monotone.
Find the limit.
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 Winter '08
 Naik
 Limits, Mathematical analysis, Limit of a sequence, Cauchy sequence, Dominated convergence theorem

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