JAWABAN

1. INTRODUCTION TO REAL ANALYSIS 1
INDIVIDUAL TASK
EXERCISES 3.1
SEQUENCE AND THEIR LIMITS
By:
Muhammad Nur Chalim
4101414101
MATHEMATICS DEPARTMENT
MATHEMATICS AND NATURAL SCIENCES FACULTY
SEMARANG STATE UNIVERSITY
2016

2. EXERCISES 3.1
Problem
1. The sequence (𝑥 𝑛) is defined by the following formulas for the 𝑛th term. Write the first
five terms in each case:
a. 𝑥 𝑛 ≔ 1 + (−1) 𝑛
b. 𝑥 𝑛 ≔
(−1) 𝑛
𝑛
c. 𝑥 𝑛: =
1
𝑛(𝑛+1)
d. 𝑥 ≔
1
𝑛2 +2
2. The first few terms of a sequence (𝑥 𝑛) are given below. Assuming that the natural
pattern indicated by these terms persists, give a formula for the 𝑛th term 𝑥 𝑛
a. 5,7,9,11,. . .
b.
𝟏
𝟐
, −
𝟏
𝟒
,
𝟏
𝟖
, −
𝟏
𝟏𝟔
, …
c.
𝟏
𝟐
,
𝟐
𝟑
,
𝟑
𝟒
,
𝟒
𝟓
, ….
d. 1,4,9,16,. . .
3. List the first five terms of the following inductively defined sequences
a. 𝑥1 ≔ 1, 𝑥 𝑛+1 = 3𝑥 + 1
b. 𝑦1; = 2, 𝑦 𝑛+1 =
1
2
(𝑦 𝑛 +
2
𝑦 𝑛
)
c. 𝑧1 ≔ 1, 𝑧2 ≔ 2, 𝑧 𝑛+2 ≔
𝑧 𝑛+1+𝑧 𝑛
𝑧 𝑛+1−𝑧 𝑛
d. 𝒔 𝟏 = 𝟑, 𝒔 𝟐 ≔ 𝟓, 𝒔 𝒏+𝟏 ≔ 𝒔 𝒏 + 𝒔 𝒏+𝟏
4. For any 𝑏 ∈ 𝑅, prove that lim (
𝑏
𝑛
) = 0
Solution
1. The sequence (𝑥 𝑛) is defined by the following formulas for the 𝑛th term. Write the first
five terms in each case:
a. 𝑥 𝑛 ≔ 1 + (−1) 𝑛
If 𝑛 = 1 then 𝑥1 ≔ 1 + (−1)1
= 0.

3. If 𝑛 = 2 then 𝑥2 ≔ 1 + (−1)2
= 2.
If 𝑛 = 3 then 𝑥3 ≔ 1 + (−1)3
= 0.
If 𝑛 = 4 then 𝑥4 ≔ 1 + (−1)4
= 2.
If 𝑛 = 5 then 𝑥5 ≔ 1 + (−1)5
= 0.
Thus, 𝑥 𝑛 = (0, 2, 0,2, 0).
b. 𝑥 𝑛 ≔
1
𝑛( 𝑛+1)
If 𝑛 = 1 then 𝑥1 ≔
1
1(1+1)
=
1
2
.
If 𝑛 = 2 then 𝑥2 ≔
1
2(2+1)
=
1
6
.
If 𝑛 = 3 then 𝑥3 ≔
1
3(3+1)
=
1
12
.
If 𝑛 = 4 then 𝑥4 ≔
1
4(4+1)
=
1
20
.
If 𝑛 = 5 then 𝑥5 ≔
1
5(5+1)
=
1
30
.
Thus, 𝑥 𝑛 = (
1
2
,
1
6
,
1
12
,
1
20
,
1
30
).
c. 𝑥 𝑛 ≔
(−1) 𝑛
𝑛
If 𝑛 = 1 then 𝑥1 ≔
(−1)1
1
= −1.
If 𝑛 = 2 then 𝑥2 ≔
(−1)2
2
=
1
2
.
If 𝑛 = 3 then 𝑥3 ≔
(−1)3
3
= −
1
3
.
If 𝑛 = 4 then 𝑥4 ≔
(−1)4
4
=
1
4
.
If 𝑛 = 5 then 𝑥5 ≔
(−1)5
5
= −
1
5
.
Thus, 𝑥 𝑛 = (−1,
1
2
, −
1
3
,
1
4
, −
1
5
).

4. d. 𝑥 𝑛 ≔
1
𝑛2+2
If 𝑛 = 1 then 𝑥1 ≔
1
12+2
=
1
3
.
If 𝑛 = 2 then 𝑥2 ≔
1
22+2
=
1
6
.
If 𝑛 = 3 then 𝑥3 ≔
1
32+2
=
1
11
.
If 𝑛 = 4 then 𝑥4 ≔
1
42 +2
=
1
18
.
If 𝑛 = 5 then 𝑥5 ≔
1
52+2
=
1
27
.
Thus, 𝑥 𝑛 = (
1
3
,
1
6
,
1
11
,
1
18
,
1
27
).
2. The first few terms of a sequence (𝑥 𝑛) are given below. Assuming that the natural
pattern indicated by these terms persists, give a formula for the 𝑛th term 𝑥 𝑛
a. 5,7,9,11,...
So 𝑥 𝑛 = 2𝑛 + 3
b.
1
2
, −
1
4
,
1
8
, −
1
16
,…
So 𝑥 𝑛 = − (−
1
2
)
𝑛
c.
1
2
,
2
3
,
3
4
,
4
5
, …
So 𝑥 𝑛 =
𝑛
𝑛+1
d. 1,4,9,16,...
So 𝑥 𝑛 = 𝑛2
3. List the first five terms of the following inductively defined sequences.
a. 𝑥1 = 1, 𝑥 𝑛+1 = 3𝑥 𝑛 + 1
Thus, 𝑥 𝑛 = (1,4, 13, 40,121)
b. 𝑦1 = 2, 𝑦 𝑛+1 =
1
2
(𝑦 𝑛 +
2
𝑦 𝑛
)
Thus, 𝑦 𝑛 = (2,
3
2
,
17
16
,
801
544
, … ,… )
c. 𝑧1 = 1, 𝑧2 = 2, 𝑧 𝑛+2 =
𝑧 𝑛+1+𝑧 𝑛
𝑧 𝑛+1−𝑧 𝑛
Thus, 𝑧 𝑛 = (1,2,3,5,4)
d. 𝑠1 = 3, 𝑠2 = 5, 𝑠 𝑛+2 = 𝑠 𝑛 + 𝑠(𝑛+1)
Thus, 𝑠 𝑛 = (3,5,8,13,21)

5. 4. For any 𝑏 ∈ 𝑅, prove that lim (
𝑏
𝑛
) = 0
Proof:
Choose 𝐾( 𝜀) >
| 𝑏|
𝜀
such that implies :
|
𝑏
𝑛
− 0| = |
𝑏
𝑛
| ≤
| 𝑏|
𝐾( 𝜀)
< 𝜀