number theory - $lim_{ntoinfty}{left({sum_{k=1}^nk^k}right)/{n^n}}=1$?

I'm interested in the following sum $S_n$.
$$S_n:=\sum_{k=1}^nk^k=1^1+2^2+3^3+\cdots+n^n.$$

Letting $T_n:={S_n}/{n^n}$, wolfram tells us the followings.
$$T_5=1.09216, T_{10}\approx1.04051, T_{30}\approx1.01263, T_{60}\approx1.00622.$$

Then, here is my expectation.

My expectation:
$$\lim_{n\to\infty}{T_n}=1.$$

It seems obvious, so I've tried to prove this, but I'm facing difficulty. Then, here is my question.

Question: Could you show me how to find $\lim_{n\to\infty}{T_n}$ if it exists?

Answer

Let $n\ge 3$. Look at the top. The sum of the terms up to and including $(n-2)^{n-2}$ is $\le (n-2)(n-2)^{n-2}$. The next term is $(n-1)^{n-1}$ and the last is of course $n^n$.

So our ratio is $\gt 1$ and less than
$$\frac{(n-2)(n-2)^{n-2}}{n^n} +\frac{(n-1)^{n-1}}{n^n}+1.\tag{1}$$
The limit of each of the first two terms of (1) is $0$. For the first term is less than $\dfrac{n\cdot n^{n-2}}{n^n}=\dfrac{1}{n}$ and the second is also $\lt \dfrac{1}{n}$.

The result now follows by Squeezing.