Ratio Test And Root Test

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MULTI VERIABLE
CALCULUS
Assignment
Group Mambers Roll No

2. Ratio Test :
Let ∑ ( 𝑎 𝑛)∞
1 be a series of positive terms hence suppose that lim
𝑛→∞
𝑎 𝑛+1
𝑎 𝑛
where L is a non negative real number or infinity.
1. If L < 1 , the series Converge.
2. If L > 1 or ∞ , the series Diverge.
3. If L = 1 , the test is fail.
Test :
lim
𝑛→∞
𝑎 𝑛 + 1
𝑎 𝑛
Question No 1:
∑ (
( 𝑛 + 2)!
4! 𝑛! 2 𝑛
)
∞
1
𝑎 𝑛 =
( 𝑛 + 2)!
4! 𝑛! 2 𝑛
𝑎 𝑛+1 =
( 𝑛 + 1 + 2)!
4! (𝑛 + 1)! 2 𝑛+1
𝑎 𝑛+1 =
( 𝑛 + 3)!
4! (𝑛 + 1)! 2 𝑛+1

3. lim
𝑛→∞
𝑎 𝑛 + 1
𝑎 𝑛
= lim
𝑛→∞
( 𝑛 + 3)!
4!(𝑛 + 1)!2 𝑛
/
( 𝑛 + 2)!
4! 𝑛!2 𝑛
= lim
𝑛→∞
( 𝑛 + 3)!
4! ( 𝑛 + 1)!2 𝑛. 2
×
4! 𝑛!2 𝑛
( 𝑛 + 2)!
= lim
𝑛→∞
( 𝑛 + 3)( 𝑛 + 2)( 𝑛 + 1)!
4! ( 𝑛 + 1) 𝑛! 2 𝑛. 2
×
4! 𝑛! 2 𝑛
( 𝑛 + 2)( 𝑛 + 1)!
= lim
𝑛→∞
( 𝑛 + 3)
( 𝑛 + 1) 2
=
1
2
lim
𝑛→∞
𝑛(1 +
3
𝑛
)
𝑛(1 +
1
𝑛
)
=
1
2
.
(1 +
3
∞
)
(1 +
1
∞
)
=
1
2
.
1 + 0
1 + 0
1
2
< 1
Converge
ROOT TEST:
Let ∑ ( 𝑎 𝑛)∞
1 be a series of positive terms hence suppose that
lim
𝑛→∞
( 𝑎 𝑛)
1
𝑛⁄
= 𝐿 where L is a non negative real number or ∞ .
If L < 1 then the series Converge.

4. If L > or ∞ then the series Diverge.
If L = 1 then the test fail.
Test :
lim
𝑛→∞
(𝑎 𝑛)
Question No 2 :
∑(
n
10
)n
∞
1
𝑎 𝑛 = (
𝑛
10
) 𝑛
lim
𝑛→∞
( 𝑎 𝑛)
1
𝑛⁄
= [(
𝑛
10
) 𝑛
]
= lim
𝑛→∞
[
𝑛
10
]
= [
∞
10
]
= ∞
Diverge