<s>[INST] Evaluate $\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$ Problem
Evaluate  $$\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$$
My solution
Notice that $$\lim_{n \to \infty}\frac{n+n^2+n^3+\cdots +n^n}{n^n}=\lim_{n \to \infty}\frac{n(n^n-1)}{(n-1)n^n}=\lim_{n \to \infty}\frac{1-\dfrac{1}{n^n}}{1-\dfrac{1}{n}}=1,$$and

$$\lim_{n \to \infty}\frac{1+2^n+3^n+\cdots+n^n}{n^n}=\frac{e}{e-1}.$$

Hence,\begin{align*}\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}&=\lim_{n \to \infty}\frac{\dfrac{n+n^2+n^3+\cdots +n^n}{n^n}}{\dfrac{1+2^n+3^n+\cdots +n^n}{n^n}}\\&=\frac{\lim\limits_{n \to \infty}\dfrac{n+n^2+n^3+\cdots +n^n}{n^n}}{\lim\limits_{n \to \infty}\dfrac{1+2^n+3^n+\cdots +n^n}{n^n}}\\&=1-\frac{1}{e}.\end{align*}
The solution posted above need to quote an uncommon limit. Is there another more simple and more direct solution?
 [/INST] A Query to @Teddy38's Answer
Yes, we can agree that $$n+n^2+\cdots+n^n=\frac{n}{n-1}(n^n-1)$$ and $$\frac{n^{n+1}}{n+1}=\int_{0}^{n}x^n\ dx<1^n+2^n+\cdots+n^n<\int_{1}^{n+1}x^n\ dx=\frac{(n+1)^{n+1}-1}{n+1}.$$ Thus,we obtain $$\frac{n+1}{n-1}\cdot \frac{n^{n+1}-n}{(n+1)^{n+1}-1}<\dfrac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}<\frac{n+1}{n-1}\cdot\frac{n^n-1}{n^n}.$$Now, let's evaluate the limits of the both sides as $n \to \infty.$ For the left side, we have $$\lim_{n \to \infty} \dfrac{n+1}{n-1} \cdot \lim_{n \to \infty}\dfrac{1-\dfrac{1}{n^n}}{\left(1+\dfrac{1}{n}\right)^n\cdot\left(1+\dfrac{1}{n}\right)-\dfrac{1}{n^{n+1}}}=\frac{1}{e}.$$
As for the right side, we have $$\lim_{n \to \infty} \dfrac{n+1}{n-1} \cdot \lim_{n \to \infty}\left(1-\dfrac{1}{n^n}\right)=1.$$
From the two aspects, we may only conclude that, $$\varliminf_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}\geq\frac{1}{e},$$and $$\varlimsup_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}\leq 1.$$ That's to say, if the limit we want really exists, then $$\frac{1}{e}\leq\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n} \leq 1.$$This is true, but can't give the accurate value of the limit we want.
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