I can show that the following limit exists but I am having difficulties to find it. It is $\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$ Can someone please help me?
How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
-
0Probably. But the result is not $\int_0^1 x^x\, dx$. – 2012-06-28
5 Answers
An asymptotic expansion can be obtained as below. More terms can be included by using more terms in the expansions of $\exp$ and $\log$. $ \begin{align} \sum_{k=0}^n\frac{k^n}{n^n} &=\sum_{k=0}^n\left(1-\frac{k}{n}\right)^n\\ &=\sum_{k=0}^n\exp\left(n\log\left(1-\frac{k}{n}\right)\right)\\ &=\sum_{k=0}^{\sqrt{n}}\exp\left(n\log\left(1-\frac{k}{n}\right)\right)+O\left(ne^{-\sqrt{n}}\right)\\ &=\sum_{k=0}^{\sqrt{n}}\exp\left(-k-\frac{1}{2n}k^2+O\left(\frac{k^3}{n^2}\right)\right)+O\left(ne^{-\sqrt{n}}\right)\\ &=\sum_{k=0}^{\sqrt{n}}e^{-k}\exp\left(-\frac{1}{2n}k^2+O\left(\frac{k^3}{n^2}\right)\right)+O\left(ne^{-\sqrt{n}}\right)\\ &=\sum_{k=0}^{\sqrt{n}}e^{-k}\left(1-\frac{1}{2n}k^2+O\left(\frac{k^ 4}{n^2}\right)\right)+O\left(ne^{-\sqrt{n}}\right)\\ &=\sum_{k=0}^{\sqrt{n}}e^{-k}-\frac{1}{2n}\sum_{k=0}^{\sqrt{n}}k^2e^{-k}+O\left(\frac{1}{n^2}\right)\\ &=\frac{e}{e-1}-\frac{1}{2n}\frac{e(e+1)}{(e-1)^3}+O\left(\frac{1}{n^2}\right) \end{align} $ Several steps use $ \sum_{k=n}^\infty e^{-k}k^m=O(e^{-n}n^m) $ which decays faster than any power of $n$.
-
0@MikeSpivey: Thanks! I am a big fan of the Euler-Maclaurin Sum Formula. For polynomials, it is exact (in fact, in [this answer](http://math.stackexchange.com/a/156085), I give a condition for the EMS Formula to converge). – 2013-01-09
Finally, I have suffered this proof. Consider functions $ f_n(x)=\left(1-\frac{\lfloor x\rfloor}{n}\right)^n\chi_{[0,n+1]}(x) $ Note that $ \int\limits_{[0,+\infty)} f_n(x)d\mu(x)=\sum\limits_{k=0}^n\int\limits_{[k,k+1)}\left(1-\frac{\lfloor x\rfloor}{n}\right)^nd\mu(x)= \sum\limits_{k=0}^n\left(1-\frac{k}{n}\right)^n $ $ \lim\limits_{n\to\infty}f_n(x)=\lim\limits_{n\to\infty}\left(1-\frac{\lfloor x\rfloor}{n}\right)^n\cdot \lim\limits_{n\to\infty}\chi_{[0,n+1]}(x)=e^{\lfloor x\rfloor} $ One may check that $\{f_n:n\in\mathbb{N}\}$ is a non-decreasing sequence of non-negative functions, then using monotone convergence theorem we get $ \lim\limits_{n\to\infty}\sum\limits_{k=0}^n\left(\frac{k}{n}\right)^n= \lim\limits_{n\to\infty}\sum\limits_{k=0}^n\left(1-\frac{k}{n}\right)^n= \lim\limits_{n\to\infty}\int\limits_{[0,+\infty)} f_n(x)d\mu(x)= $ $ \int\limits_{[0,+\infty)} \lim\limits_{n\to\infty}f_n(x)d\mu(x)= \int\limits_{[0,+\infty)} e^{\lfloor x\rfloor}d\mu(x)= \sum\limits_{k=0}^\infty e^{-k}=\frac{1}{1-e^{-1}} $
-
0@Norbert Because yesterday I thought [Poisson summation formula](http://en.wikipedia.org/wiki/Poisson_summation_formula) is more powerful in this problem than Euler-Maclaurin summation formula, where $\Theta_n(t)$ is periodic function of $t$. – 2012-06-29
Just for reference: With aid of some fancy theorem, you can skip most of hard analysis. As in other answers, we begin by writing
$ \sum_{k=1}^{n} \left( \frac{k}{n}\right)^n \ \overset{k \to n-k}{=} \ \sum_{k=0}^{n-1} \left( 1 - \frac{k}{n}\right)^n \ = \ \sum_{k=0}^{\infty} \left( 1 - \frac{k}{n}\right)^n \mathbf{1}_{\{k < n\}}, $
where $\mathbf{1}_{\{k < n\}}$ is the indicator function which takes value $1$ if $k < n$ and $0$ otherwise. Now for each $0 \leq k < n$, utilizing the inequality $\log(1-x) \leq -x$ which holds for all $x \in [0,1)$ shows that
$ \left( 1 - \frac{k}{n}\right)^n = e^{n \log(1 - \frac{k}{n})} \leq e^{-k}. $
Since $\sum_{k=0}^{\infty} e^{-k} < \infty$, by the dominated convergence theorem we can interchange the infinite sum and the limit:
$ \lim_{n\to\infty} \sum_{k=1}^{n} \left( \frac{k}{n}\right)^n = \sum_{k=0}^{\infty} \lim_{n\to\infty} \left( 1 - \frac{k}{n}\right)^n \mathbf{1}_{\{k < n\}} = \sum_{k=0}^{\infty} e^{-k} = \frac{1}{1 - e^{-1}}. $
-
0@user372003, I added a bit of details to my answer. But basically, I replaced the index $k$ by $n-k$. – 2017-09-27
Let's notice a few things. All the terms are positive, bounded between $0$ and $1$, and there is a term that is exactly $1$. What about the next largest term?
So we ask ourselves what $\lim \limits_{n \to \infty} \left( \dfrac{n-1}{n} \right)^n$ is, and after a little calculation we see that this limit is $1/e$. The 'next' term involves $\lim \limits_{n \to \infty} \left( \dfrac{n-2}{n} \right)^n = e^{-2}$. So heuristically, we would expect the limit to be
$1 + e^{-1} + e^{-2} + \dots = \frac{1}{1-\frac{1}{e}}$
Working only a little harder, you can justify that this is the limit.
-
0Yes, it's a matter of passing to the limit inside a series. – 2012-06-28
$\sum_{k=1}^n(k/n)^n=\sum_{0
Can anybody give a more accurate approximation? The key to the approximation is to find the asymptotics for $\sum_{k>0}\exp(-k-k^2/2n)$, like the Bell sum $\sum_{k>0}e^{-k^2/n}$.
Edit anon pointed out that it's theta function: $\sum_ke^{-(k+t)^2/n}$, so the Fourier series works pretty well for the asymptotics: $\Theta_n(t)=\sqrt{\pi n}\left(1+2e^{-\pi^2 n}(\cos2\pi t)+2e^{-4\pi^2 n}(\cos4\pi t)+2e^{-9\pi^2 n}(\cos6\pi t)+\cdots\right)$ But I have no idea about Fourier series because I know very little about calculus!
-
0@robjohn Thanks. – 2012-06-29