Which number is larger?

Question

If $n$ is large enough, which is greater:

$(n+1) ^{n+1}$ or $(kn)^{n}$ where $k$ is a natural number?

I've plotted a graph which suggests that the second is larger, but surely the larger power should dominate in the end?

Answer 1

dividing both sides by $n^{n+1}$ gives

$\left(1+{1\over n}\right)^{n+1}$ and ${k^n\over n}$

As $n\to\infty$ the first quantity converges to $e$, but the second goes off to infinity unless $k=1$, in which case it is already trivial that $(n+1)^{n+1}$ is the larger of the two.

Answer 2

previous clues are good enough for you.

Nevertheless, if you want to be more formal, you can use Newton's binomial expansion for the power $n + 1$, in which case you can compare the first one to $(kn)^n = k^n \times n^n \ldots$