Question

Need help with radius of convergence for series:

Answer 1

Find the radius of convergence of the power series, where c>0 and k is a positive integer
\[\sum_{0}^{\infty}\frac{ (n!)^k*x^n }{ (kn)! }\]

Answer 2

Have you considered the Ratio Test?

Answer 3

yes, but I get stuck at a particular step, let me show you

Answer 4

I apply the ratio test, and simplfy what I can and then I reach this point:
\[|x|\lim_{n \rightarrow \infty} \frac{ (n+1)^k * kn! }{ (k+kn)!}\]

Answer 5

this is where I am stuck, how do i simplfy this

Answer 6

That's very good to get that far. Now what?

\((k + kn)! = \underbrace{(k+kn)(k+kn - 1)(k+kn - 2)...(k+kn - (k-1))}_{k\;terms}[(kn)!]
\)

Does that get us anywhere? There are k factors of (n+1) in the numerator. Can we draw a useful conclusion?

Answer 7

\[\lim\frac{ (n+1)!)^k*x^{n+1} }{ (k(n+1))! }\frac{ (kn)! }{ (n!)^k*x^n }\]
\[|x|~\lim \frac{ (n+1)^k}{ (k(n+1))! }\frac{ (kn)! }{1}\]
the n+1 exponential always gets beaten up by the factorials
\[|x|~\lim \frac{ 1}{ (k(n+1))! }\frac{ (kn)! }{1}\]
and the bottom overpowers the top

Answer 8

* Of course, I meant "k factors", but I'm not typing all that again.

Answer 9

lol, but you have to, otherwise its just senseless :)

Answer 10

(50*500)!
---------
(50*501)!

1/
8301079125393033910740504709928984018467750442134753707306295783809433798515594730033373466585741566560097896357676899850578001105929021205193914475181694952318094683693076365798505410485299941508842113728512000000000000

Answer 11

is k an integer? or can it go fraction?

Answer 12

why would they message for help and then just not be here .....

Answer 13

Note: I'm nervous about the problem statement suggesting that c > 0. "c"? Where is that?

Answer 14

hmm, i spose reading the question might be prudent :) k is a positive integer

Answer 15

convergence is greater than zero? dunno