MATH 147 Assignment #2
Due: Friday, October 9
1)
For each of the following sequences determine if it converges or diverges.
If it converges ﬁnd the limit:
a)
{
n
n
+1
}
b)
{
sin(
n
)
n
}
c)
{
√
n
+ 1

√
n
}
d)
e
n
n
!
e)
{
n
cos(
nπ
)
2
n
+1
}
2)
We will later be able to show that
1
n
+ 1
<
ln(
n
+ 1)

ln(
n
)
<
1
n
a)
Let
a
n
= 1 +
1
2
+
1
3
+
···
+
1
n

ln(
n
). Prove that
{
a
n
}
converges.
Note:
γ
= lim
n
→∞
a
n
is called Euler’s constant. It is still not known
whether or not
γ
is rational.
b)
Show that if
b
n
= 1 +
1
2
+
1
3
+
···
+
1
n
, then ln(
n
+ 1)
< b
n
≤
ln(
n
) + 1.
c)
Estimate how many terms it would take before
b
n
>
10
6
. How long do
you think it would it take for a modern computer to perform this many
additions?
3) a)
Suppose that
a
n
≥
0 and that lim
n
→∞
a
n
=
L
. Show that lim
n
→∞
√
a
n
=
√
L
.
(Hint: Do the cases
L
= 0 and
L >
0 separately. When
L >
0 show
that
√
a
n

√
L
=
a
n

L
√
a
n
+
√
L
.
b)
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 Fall '09
 Wolzcuk
 Math, Topology, lim, Empty set, Closed set, bn ≤ ln

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