Convergence of (n!)^2 / (kn)! for Positive Integers k: Ratio Test Explanation

[remaan]

The hardest ever,,,

Homework Statement

40) for which positive intigers k is the following sries convergent:
(n!) ^2 / (kn)!

Homework Equations

using the ratio test the limit should be less than 1

The Attempt at a Solution

I tried the ratio test, but I got inifinty?

 

[tiny-tim]

Hi remaan!

for which positive intigers k is the following sries convergent:
(n!) ^2 / (kn)!
…
using the ratio test the limit should be less than 1
…
I tried the ratio test, but I got inifinty?

Hint: which is larger, (5!)² or 10! ?

 

[Hurkyl]

using the ratio test the limit should be less than 1

No -- it could be less than 1, but it doesn't have to. You're making some assumptions when you said that; what are they? (Or, I suppose you could have just made a mistake)

I tried the ratio test, but I got inifinty?

What does that tell you?

 

[remaan]

Hi remaan!


Hint: which is larger, (5!)² or 10! ?

Hi,

mm, 1o! is greater, but how can I benifit from this in finding K ??

 

[remaan]

No -- it could be less than 1, but it doesn't have to. You're making some assumptions when you said that; what are they? (Or, I suppose you could have just made a mistake)

I mention that because the question says "positive intergers,"


What does that tell you?

It tells that the series is not alternating and I can't ( ratio test ) to find the k .

 

[Hurkyl]

I can't ( ratio test ) to find the k .

Why not? The only time the ratio test is inconclusive is when the limit is 1, or doesn't exist, and for all possible values of k, you're not in either of those cases