convergence with factorial/multiplied factorial

[sambellamy]

I have to determine for which positive integers k the series will be convergent:
Ʃ_n=1^∞ (n!)² / (kn)!

I know that k=1 will cause the values to grow without bound and diverge. I also have determined that k = 2 oranything greater will cause the denominator to grow more than the numerator and the values will converge on zero. I am not sure how to approach this mathematically to show that it is ture. I tried with the ratio test and the example that k = 2, but none of the terms I got could cancel another out. What does (kn)! break down into? I know that k(n!) and kⁿ(n!) are too small compared to (kn)!, and so is k!n!. Please help!

 

[stapel]

I have to determine for which positive integers k the series will be convergent:
Ʃ_n=1^∞ (n!)² / (kn)!

... I tried with the ratio test..., but none of the terms I got could cancel another out.

Please reply showing what you did and what you got. You set up the ratio:

. . . . .\(\displaystyle \displaystyle{ \frac{a_{n+1}}{a_n}\, =\, \frac{\left(\frac{\left((n\, +\, 1)!\right)^2}{\left(k(n\, +\, 1)\right)!}\right)} {\left(\frac{(n!)^2}{(kn)!}\right)} \, =\, \frac{\left(\frac{\left((n\, +\, 1)(n!)\right)^2}{(kn\, +\, k)!}\right)}{\left(\frac{(n!)^2}{(kn)!}\right)} }\)

Then what?

 

[sambellamy]

I have retried this and been able to whittle down what was posted above (I don't know how to write equations in the comments) to the following:

|(n+1)² / (kn+1)|

as n approaches infinity, the limit of the sum will also grow without bound. i have no idea what this tells me about the values of k. do i need to make kn+1 greater than (n+1)² ? should i follow the ratio test through, and say that this absolutely diverges? is the ratio test any help at all? are my initial assumptions about the k values correct?

 

[stapel]

I have retried this and been able to whittle down what was posted above (I don't know how to write equations in the comments) to the following:

|(n+1)² / (kn+1)|

How did you get this denominator? You had (kn)! over (kn + k)! = [(kn)!](kn + 1)(kn + 2)...(kn + (k-1))(kn + k). What happened to all the other terms?

Please show your steps. Thank you!

 

[sambellamy]

This is what I did:

|((n+1)!)² / (kn+1)! *((kn)! / (n!)² |

|(n+1)² (n!)² / (kn+1)! * (kn)! | i cancelled the n! terms

|(n+1)² / (kn+1)! * (kn)! | i assumed that (kn+1)! = (kn)!(kn+!) and cancelled those:

|(n+1)² / (kn+1)|

and i have lim as n --> inf. 2n+2 / k

where did i go wrong?

Thanks!

 

[sambellamy]

I have revisited this a bit and realised that the a_n+1 morphing of (kn)! should be (kn+k)! instead of (kn+1)!. I do not think that (kn+k)! can be broken down into (kn)!*(kn+k), because there may be integers between kn and kn+k that will not be accounted for. Therefore, I have:

|((n+1)!)² / (kn+k)! *((kn)! / (n!)² |

|(n+1)² (n!)² / (kn+k)! * (kn)! | i cancelled the n! terms

|(n+1)²*(kn)! / (kn+k)!|

any further advice?

 

[stapel]

This is what I did:

|((n+1)!)² / (kn+1)! *((kn)! / (n!)² |

Okay; again: where did the "(kn+1)!" come from? You'd started with (k(n+1))! = (kn+k)!. How did this become (kn+1)!? What happened with all the other terms (listed in my previous reply)? What happened between what I posted (which is where you should have started) and what you've posted (which is necessarily quite a few steps away)?

 

[sambellamy]

I think we posted at the same time. I appreciate your helping me with this problem!

I unpacked (kn+k)! and decided that it was equal to (kn)!*k. so i had:

| (n+1)² (kn)! / (kn)!*k | which, upon cancelling the (kn)! term, became:

|(n+1)² / k|

|1/k|(n+1)²

I think this is correct, but I'm still not sure how it helps me prove that when k is 1, the series converges, and when it is larger the series diverges.

 

[pka]

I think we posted at the same time. I appreciate your helping me with this problem!

I unpacked (kn+k)! and decided that it was equal to (kn)!*k. so i had.

You best rethink that.
\(\displaystyle [(3\cdot 4+3)!=15!\\(3\cdot 4)!\cdot 3=12!\cdot 3\)