Question

Limits

Answer 1

\[\Huge \lim_{n \rightarrow \infty} \left\{ \frac{n!}{(kn)^n} \right\}^\frac{1}{n}\]

Answer 2

Can someone get this in the form of
\(\Huge \sum_{r=1}^{n} \frac{r}{n}\) form? I can do the rest.

Answer 3

Let the limit \(=L\), then take the \(\ln\) of both sides.

Answer 4

dont think that would work?

Answer 5

hmmm..ookay proceed though

Answer 6

So just focusing on the inside part: \[
\ln\left[\left(\frac{n!}{(kn)^n}\right)^{1/n}\right]
\]

Answer 7

\[
\frac 1n\ln(n!) - \frac 1n\ln((kn)^n)
\]

Answer 8

okay

Answer 9

\[
\frac 1n\ln(n!)-(\ln(k)+\ln(n))
\]

Answer 10

\[
\ln(n!)=\sum_{k=1}^n\ln(k)
\]

Answer 11

There is the summation you're looking for.