Stuck on a series question using root or ratio text

[kluda06]

The problem is

Σ <- infinity, n=1 (on the bottom)

Σ (n!)ⁿ/(nⁿ)² converges to:

its either
(1/2)
0
1
2
diverges to infinity

I first re wrote the problem as, (n!)ⁿ/(n²)ⁿ => (n!/n²)ⁿ

not sure if that was even right for me to do. but when i put lim u_k+1/u_k ​it got real confusing. Help me out please!

 

[DrPhil]

The problem is

Σ <- infinity, n=1 (on the bottom)

Σ (n!)ⁿ/(nⁿ)² converges to:

its either
(1/2)
0
1
2
diverges to infinity

I first re wrote the problem as, (n!)ⁿ/(n²)ⁿ => (n!/n²)ⁿ OK
not sure if that was even right for me to do. but when i put lim u_k+1/u_k ​it got real confusing. Help me out please!

Sometimes it is helpful to write out the first few terms.

\(\displaystyle \displaystyle \sum_{n=1}^\infty \left(\dfrac{n!}{n^2}\right)^n = \sum_{n=1}^\infty \left(\dfrac{(n-1)!}{n}\right)^n\)

................\(\displaystyle = 1 + \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{2}{3}\right)^3 + \left(\dfrac{6}{4}\right)^4 + \left(\dfrac{24}{5}\right)^5 + \ \cdot \ \cdot \cdot\)

Pretty clear what the result is.. what test would you use?

 

[pka]

The problem is
Σ (n!)ⁿ/(nⁿ)² converges to:

I first re wrote the problem as, (n!)ⁿ/(n²)ⁿ => (n!/n²)ⁿ

Use the root test.

\(\displaystyle \displaystyle{\lim _{n \to \infty }}\frac{{n!}}{{{n^2}}} = {\lim _{n \to \infty }}\left( {\frac{{n - 1}}{n}} \right)\left( {n - 2} \right)!=?\)

 

[kluda06]

how did you get it to (n-1)? i did plug in 1,2,3, into the problem and got the same sequence but i want to know how did you get the (n-1)!/n

Sometimes it is helpful to write out the first few terms.

\(\displaystyle \displaystyle \sum_{n=1}^\infty \left(\dfrac{n!}{n^2}\right)^n = \sum_{n=1}^\infty \left(\dfrac{(n-1)!}{n}\right)^n\)

................\(\displaystyle = 1 + \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{2}{3}\right)^3 + \left(\dfrac{6}{4}\right)^4 + \left(\dfrac{24}{5}\right)^5 + \ \cdot \ \cdot \cdot\)

Pretty clear what the result is.. what test would you use?

 

[DrPhil]

how did you get it to (n-1)? i did plug in 1,2,3, into the problem and got the same sequence but i want to know how did you get the (n-1)!/n

One power of n cancels:

\(\displaystyle \displaystyle \dfrac{n!}{n^2} = \dfrac{n \times (n-1)!}{n \times n} = \dfrac{(n-1)!}{n}\)