3c) S
1
=2
≥
1
Assume k
∈
S for S
k+1
≥
S
k
Then S
1
=2, S
2
=2.75, S
3
=3.125, S
4
=3.3125
The sequence is monotone because it is always increasing for each N
∈ℕ
. The monotone
sequence is bounded above, and converges to 3.5. The limit of the sequence is 3.5.
4a) a
n
=
(−1)
𝑛
𝑛
, { -1, ½, -
1/3, ¼, …}
4b) a
n
= n or a
n
= -n
4c) a
n
=
(−1)
𝑛
,
∀
n
∈ℕ
is bounded from -1 to 1 but does not converge. It has two limits as n
approaches
+
∞
6a) Suppose a
n+1
= a
n
+2
, ∀
n
∈ℕ
.
a
1
=2, a
2
=3, a
3
=5, a
4
=7. The sequence is monotone because it is
increasing and a
n
is a multiple of n, for some k
∈ ℝ
,, then a
n
is increasing with k
c
n
is monotone
for k
∈ ℝ
6b) Suppose a
n+1
= a
n
+2 be an increasing sequence because a
1
=1 and a
n+1
≥
a
n
, for any n
∈ℕ
.

Let
b
n
=
𝑛+1
𝑛
2
for any n
∈ℕ
.
b
1
=2, b
2
=4, b
3
=
4
9
, b
4
=
5
16
. The sequence is a decreasing sequence.
If C
n=
?
𝑛
?
𝑛
, then C
1
=1/2, C
2
= 4, C
3
=11.25, and C
4
=22.4. C
n
is strictly increasing
monotone.
Similarly if a
n
=
𝑛+1
𝑛
2
and b
n+1
= b
n
+2, then C
n
is strictly decreasing and is monotone.
9) We can begin by squaring to get rid of the square root, then we have:
(S
n+1
)
2
-x=
(
𝑆
𝑛
2
+𝑥
2𝑆
𝑛
)
2
-x
𝑆
𝑛
4
+2𝑆
𝑛
𝑥+𝑥
2
−4𝑆
𝑛
2
𝑥
4𝑆
𝑛
2
𝑆
𝑛
4
−2𝑆
𝑛
𝑥+𝑥
2
4𝑆
𝑛
2
(
𝑆
𝑛
2
−𝑥
2𝑆
𝑛
)
2
≥
0
So S
n
≥ √𝑥
for any n
≥
2, so S
n
is bounded below by
√𝑥
.

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- Fall '08
- Staff
- Tn, Sn, lim sn