testing a series involving factorials

[njmiano]

I am trying to decide whether the following converges or diverges.
(Sum from n =1 to infinity) (n!)^n / n^4n

I have not had much experience working with factorials, so when I try to find this with the Ratio Test, I get lost.
If someone could walk me through this problem, I would be very grateful.

 

[Deleted member 4993]

I am trying to decide whether the following converges or diverges.
(Sum from n =1 to infinity) (n!)^n / n^4n

I have not had much experience working with factorials, so when I try to find this with the Ratio Test, I get lost.
If someone could walk me through this problem, I would be very grateful.

Please show your work, even if you are lost - so that we know where to begin to help you.

 

[njmiano]

All I know is to set up the problem in the following way and try to test if L is <,>,= 1
L = ((n+1)! ^n+1 / n^4n+1) * (n^4n / (n!)^n)
from there I get lost.

 

[Deleted member 4993]

All I know is to set up the problem in the following way and try to test if L is <,>,= 1
L = ((n+1)! ^n+1 / n^4n+1) * (n^4n / (n!)^n)
from there I get lost.

\(\displaystyle \frac{(n+1)!^{n+1}}{(n+1)^{4(n+1)}}\cdot\frac{n^{4n}}{(n!)^n}\)

\(\displaystyle \frac{(n+1)!^n\cdot (n+1)!}{(n+1)^{4n}\cdot (n+1)^4}}\cdot\frac{n^{4n}}{(n!)^n}\)

\(\displaystyle \frac{(n)!^n\cdot (n+1)^n \cdot (n+1)\cdot n!}{(n+1)^{4n}\cdot (n+1)^4}}\cdot\frac{n^{4n}}{(n!)^n}\)

Now continue - very carefully ......

 

[Deleted member 4993]

Another test would be just check a[sub:137eag6l]n[/sub:137eag6l]

\(\displaystyle a_n \, = \frac{(n!)^n}{n^{4n}}\)

\(\displaystyle = \, [\frac{n!}{n^4}]^n\)

\(\displaystyle = \, [\frac{n\cdot (n-1) \cdot (n-2) \cdot (n-3) \cdot (n-4)!}{n^4}]^n\)

\(\displaystyle = \, [\frac{n}{n}\cdot \frac{(n-1)}{n} \cdot \frac{(n-2)}{n} \cdot \frac{(n-3)}{n} \cdot (n-4)!]^n\)

Now get the limit....