Power Series - Interval of Convergence Problem

[mcdowellmg]

Homework Statement

For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.)

Summation of n=1 to infinity of (n!)^2 / (kn)!

Homework Equations

ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the original equation's n's replaced with n+1 and then divide that by the original equation)

The Attempt at a Solution

I am getting lost in how to simplify everything in order to find a convergence (or not). I have limit n --> infinity of [((n_1)!)^2 / (kn+1)!] * (kn)!/(n!)^2. Basically, I am multiplying by the reciprocal. I turned (kn+1)! into (kn+1)(kn!) and canceled out the other (kn)!, and I know I need to do something similar to (n!)^2, but I am not sure what to do?

Thanks!

 

[Mark44]

Homework Statement

For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.)

Summation of n=1 to infinity of (n!)^2 / (kn)!

Homework Equations

ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the original equation's n's replaced with n+1 and then divide that by the original equation)

These are expressions you're working with, not equations.

The Attempt at a Solution

I am getting lost in how to simplify everything in order to find a convergence (or not). I have limit n --> infinity of [((n_1)!)^2 / (kn+1)!] * (kn)!/(n!)^2. Basically, I am multiplying by the reciprocal. I turned (kn+1)! into (kn+1)(kn!) and canceled out the other (kn)!, and I know I need to do something similar to (n!)^2, but I am not sure what to do?

Using the ratio test you should be working with the following limit:
\lim_{n \to \infty}\frac{a_{n + 1}}{a_n}= \lim_{n \to \infty}\frac{[(n+1)!]^2}{[k(n + 1)]!} \frac{(kn)!}{(n!)^2}

The key to simplification of these expressions is recognizing that [(n + 1)!]² = (n + 1)² * n² * (n - 1)² * ... * 3² * 2² = (n + 1)² * (n!)².

Similarly, [k(n + 1)]! = (kn + k) * (kn + k - 1)* (kn * k - 2) * ... * (kn + 1) * (kn)!.

 

[mcdowellmg]

Thank you!

I ended up getting [2,∞) with your help.

 

[Mark44]

Then I think you made a mistake.

 

[mcdowellmg]

Well, now I am confused because it was listed as correct on WebAssign, the homework website I am using for my class.

 

[Mark44]

I take back what I said. I mistakenly expanded [k(n + 1)]! into k(n+1) * (kn)! It should be [k(n + 1)]! = (kn + k)(kn + k - 1)(kn + k - 2)* ... *(kn + 1) * (kn)!

I have edited my earlier post.