1. In the name of Allah
the most gracious, the most merciful

2. MTH3101
TUTORIAL 9

3. Convergency
can be tested by
Comparison test 1
and test 2
1
𝑛2
𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠
1
𝑛
𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠 1
𝑛
𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠
1
𝑛2
𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠
𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠𝑠𝑒𝑟𝑖𝑒𝑠
Ratio Test
lim
𝑛→∞
𝑈 𝑛+1
𝑈 𝑛
= 𝐿
𝐿
𝑈 𝑛
0 < 𝐿 < 1 converges
𝐿 > 1 diverges
𝐿 = 1 Test fails
Root Test
lim
𝑛→∞
𝑈 𝑛
𝑛
= 𝑝
𝑝
𝑈 𝑛
𝑝 < 1 converges
𝑝 > 1 diverges
Integral Test
𝑓 𝑥 𝑑𝑥
∞
1
𝑓 𝑥 𝑑𝑥
∞
1
𝑈 𝑛
converges converges
diverges diverges

4. @
Check whether the series are converge or not.
𝑛 → ∞,
1
𝑛
→0
3𝑛 + 1
4𝑛3 + 𝑛2 − 2
>
3𝑛
4𝑛3 + 𝑛2 − 2
>
3𝑛
4𝑛3 + 𝑛2 >
3
4𝑛2 + 𝑛
𝑆𝑖𝑛𝑐𝑒
1
𝑛2 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠 , 𝑠𝑜
3𝑛 + 1
4𝑛3 + 𝑛2 − 2
𝑎𝑙𝑠𝑜 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠.
1.
3𝑛 + 1
4𝑛3 + 𝑛2 − 2
𝐶𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛
𝑇𝑒𝑠𝑡 1
𝐶𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛
𝑇𝑒𝑠𝑡 2
L𝑒𝑡 𝑈 𝑛 =
3𝑛 + 1
4𝑛3 + 𝑛2 − 2
, 𝑉𝑛 =
1
𝑛2
(𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠)
lim
𝑛→∞
𝑉𝑛
𝑈 𝑛
=
4𝑛3 + 𝑛2 − 2
3𝑛3 + 𝑛2 =
4
𝑛3
𝑛3 +
𝑛2
𝑛3 −
2
𝑛3
3
𝑛3
𝑛3 +
𝑛2
𝑛3
=
4
3
> 0
𝑆𝑖𝑛𝑐𝑒
1
𝑛2
𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠 , 𝑠𝑜
3𝑛 + 1
4𝑛3 + 𝑛2 − 2
𝑎𝑙𝑠𝑜 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟.

5. 𝑛 → ∞,
1
𝑛
→0
Check whether the series are converge or not.
2.
𝑛2
𝑒 𝑛3 Kalau power banyak, guna root test
L𝑒𝑡 𝑈 𝑛 =
𝑛2
𝑒 𝑛3 ,
Root Test
lim
𝑛→∞
𝑈 𝑛
𝑛
= lim
𝑛→∞
𝑛2
𝑒 𝑛3 =
𝑛
lim
𝑛→∞
𝑛2
𝑒 𝑛3
1
𝑛
= lim
𝑛→∞
𝑛
2
𝑛
𝑒 𝑛2 = 0 < 1
𝑝
𝑈 𝑛
𝑝 < 1 converges
𝑝 > 1 diverges
∴ 𝑈 𝑛 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠.
Root Test
lim
𝑛→∞
𝑈 𝑛
𝑛
= 𝑝

6. Check whether the series are converge or not.
3.
23𝑛
32𝑛 Kalau power banyak, guna root test
L𝑒𝑡 𝑈 𝑛 =
23𝑛
32𝑛 ,
Root Test
lim
𝑛→∞
𝑈 𝑛
𝑛
= lim
𝑛→∞
23𝑛
32𝑛
=
𝑛
lim
𝑛→∞
23𝑛
32𝑛
1
𝑛
= lim
𝑛→∞
23
32
=
8
9
< 1
𝑝
𝑈 𝑛
𝑝 < 1 converges
𝑝 > 1 diverges
∴ 𝑈 𝑛 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠.
Root Test
lim
𝑛→∞
𝑈 𝑛
𝑛
= 𝑝

7. Check whether the series are converge or not.
4.
100 𝑛
𝑛! Kalau ada tanda seru (factorial), selalu
guna ratio test
L𝑒𝑡𝑈 𝑛 =
100 𝑛
𝑛!
Ratio Test
∴ 𝑈 𝑛 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠.
lim
𝑛→∞
𝑈 𝑛+1
𝑈 𝑛
= lim
𝑛→∞
100 𝑛+1
(𝑛 + 1)!
100 𝑛
𝑛!
= lim
𝑛→∞
100 𝑛+1
(𝑛 + 1)!
×
𝑛!
100 𝑛
= lim
𝑛→∞
100 𝑛
× 100 × 𝑛!
100 𝑛 × (𝑛 + 1)!
= lim
𝑛→∞
100
𝑛 + 1
= 0 < 1
𝐿
𝑈 𝑛
0 < 𝐿 < 1 converges
𝐿 > 1 diverges
𝐿 = 1 Test fails
𝑛!
𝑛 + 1 !
=
1
𝑛 + 1
𝑒𝑥:
3!
3 + 1 !
=
3!
4!
=
3 × 2 × 1
4 × 3 × 2 × 1
=
1
4
=
1
3 + 1
𝑛 → ∞,
1
𝑛
→0

8. Show
9
8
<
1
𝑛3
<
5
4
∞
𝑛=1
.
Show
9
8
<
1
𝑛3
∞
𝑛=1
. Show
1
𝑛3
<
5
4
∞
𝑛=1
.
1
𝑛3
∞
𝑛=1
= 1 +
1
23
+
1
33
+
1
43
+ ⋯
=
9
8
+
1
27
+ ⋯ >
9
8
∴
1
𝑛3
∞
𝑛=1
>
9
8
1
𝑛3
∞
𝑛=1
= 1 +
1
23 +
1
33 +
1
43 + ⋯
1
𝑛3 <
1
𝑘3 + 𝑓 𝑥 𝑑𝑥
∞
𝑘
∞
𝑛=1
.
1 +
1
𝑛3
+ ⋯ < 1 +
1
𝑘3
+ ⋯ +
1
2𝑘2
1 +
1
23
+ ⋯ < 1 +
1
23
+ ⋯ +
1
2 × 22
𝑤ℎ𝑒𝑛 𝑛 = 1, 𝑘 = 1
∴
1
𝑛3 <
5
4
∞
𝑛=1
.
1
𝑛3 < 1 +
1
8
+
1
8
∞
𝑛=1
Integral Test

9. • Thank you, Good luck.
• Banyakkan latihan. Moga Berjaya. Ingat Allah
selalu.