MHB Prove the lim n→∞ (1^1+2^2+3^3+....+(n−1)^(n−1)+n^n)/(n^n)=1.

[lfdahl]

Prove

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

 

[MountEvariste]

Spoiler

Call the limit $\ell$. Obviously $1 \leqslant \ell $. For any $n,k \in \mathbb{N}\setminus\left\{0\right\}$ s.t. $k \leqslant n$ we have $k^k \leqslant n^k$, so we have:

$\displaystyle 1 \leqslant \ell \leqslant \lim_{{n}\to{\infty}}\frac{n^1+n^2+n^3+...+(n-1)^{n-1}+n^n}{n^n} = \lim_{n \to \infty} \frac{1}{n^n} \cdot \frac{n^{n+1}-n}{n-1} =\lim_{n \to \infty} \frac{1- {1}/{n^{n}}}{1-1/n} = 1. $

Thus $\ell = 1$. (First equality is geometric series). I'd love to know if it can be done via Riemann sums.

 

[lfdahl]

Spoiler

Call the limit $\ell$. Obviously $1 \leqslant \ell $. For any $n,k \in \mathbb{N}\setminus\left\{0\right\}$ s.t. $k \leqslant n$ we have $k^k \leqslant n^k$, so we have:

$\displaystyle 1 \leqslant \ell \leqslant \lim_{{n}\to{\infty}}\frac{n^1+n^2+n^3+...+(n-1)^{n-1}+n^n}{n^n} = \lim_{n \to \infty} \frac{1}{n^n} \cdot \frac{n^{n+1}-n}{n-1} =\lim_{n \to \infty} \frac{1- {1}/{n^{n}}}{1-1/n} = 1. $

Thus $\ell = 1$. (First equality is geometric series). I'd love to know if it can be done via Riemann sums.

Thankyou, June29, for your participation. Well done!(Nod)

Maybe, we should continue this thread, and ask in the forum, if the limit can be found by means of Riemanns sums?

 

[MountEvariste]

Thankyou, June29, for your participation. Well done!(Nod)

Maybe, we should continue this thread, and ask in the forum, if the limit can be found by means of Riemanns sums?

Thanks. I've finally got around to ask that question.

See https://mathhelpboards.com/calculus-10/sum-powers-limit-via-riemann-sums-23664.html