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Determine whether the following series converges. Justify your answer9(4k)!Σk=1 (kl)*Select the correct choice below and fill in the answer box to complete your choice.(Type an exact answer.)O A. The series is a qeometric series with common ratioless than 1, so the series converges by the properties of a geometric series.ThisO B. The Ratio Test yields rThis is less than 1, so the series converges by the Ratio Test.O C. The series is a geometric series with common ratioThisgreater than 1, so the series diverges by the properties of a geometric series.O D. The Ratio Test vields r=.This is greater than 1,so the series diverges by the Ratio TestO E. The limit of the terms of the series isso the series converges by the Divergence Test.8

Question
Determine whether the following series converges. Justify your answer
9(4k)!
Σ
k=1 (kl)*
Select the correct choice below and fill in the answer box to complete your choice.
(Type an exact answer.)
O A. The series is a qeometric series with common ratio
less than 1, so the series converges by the properties of a geometric series.
This
O B. The Ratio Test yields r
This is less than 1, so the series converges by the Ratio Test.
O C. The series is a geometric series with common ratio
This
greater than 1, so the series diverges by the properties of a geometric series.
O D. The Ratio Test vields r=.This is greater than 1,so the series diverges by the Ratio Test
O E. The limit of the terms of the series is
so the series converges by the Divergence Test.
8
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Determine whether the following series converges. Justify your answer 9(4k)! Σ k=1 (kl)* Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. The series is a qeometric series with common ratio less than 1, so the series converges by the properties of a geometric series. This O B. The Ratio Test yields r This is less than 1, so the series converges by the Ratio Test. O C. The series is a geometric series with common ratio This greater than 1, so the series diverges by the properties of a geometric series. O D. The Ratio Test vields r=.This is greater than 1,so the series diverges by the Ratio Test O E. The limit of the terms of the series is so the series converges by the Divergence Test. 8

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Step 1

Consider the given series:

9(4k)!
Σ
k=1 (k!)
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9(4k)! Σ k=1 (k!)

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Step 2

Now, let

9(4k
)!
4
(k!
)
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9(4k )! 4 (k! )

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Step 3

Now, apply ratio test:

First evalua...

9(4(k+1
k+1
k- 9(4k)!
r=lim "k+1= lim
4
(!)
4
(k!)
(4k+4
= lim
(4k(k+1)!
4
(4k+4)(4k+3)(4k+2)(4k+1)(4k)!
- lim
ko
(k!)
(k+1)(k!)
(4k)!
(4k+4)(4k+3)(4k+2)(4k+1)
lim
ko
4
(k+1)
4
4+
+
3
2
4+
k
1
4+
k
lim
ko
4
1
1+
4) (4)(4) (4)
=
-256
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9(4(k+1 k+1 k- 9(4k)! r=lim "k+1= lim 4 (!) 4 (k!) (4k+4 = lim (4k(k+1)! 4 (4k+4)(4k+3)(4k+2)(4k+1)(4k)! - lim ko (k!) (k+1)(k!) (4k)! (4k+4)(4k+3)(4k+2)(4k+1) lim ko 4 (k+1) 4 4+ + 3 2 4+ k 1 4+ k lim ko 4 1 1+ 4) (4)(4) (4) = -256

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