# A limit on binomial coefficients

Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.

What I can do is just use Stolz formula. But I could not proceed.

The limit is $\frac{1}{2}$. We have $$\begin{eqnarray*} \sum_{k=0}^n \log {n\choose k} &=& \sum_{k=0}^n \log n! - \sum_{k=0}^n \log k! - \sum_{k=0}^n \log (n-k)! \\ &=& (n+1)\log n! - 2\sum_{k=1}^n \log k!. \end{eqnarray*}$$ But $$\begin{eqnarray*} \sum_{k=1}^n \log k! &=& \sum_{k=1}^n \sum_{j=1}^k \log j \\ &=& \sum_{j=1}^n \sum_{k=j}^n \log j \\ &=& \sum_{j=1}^n (n-j+1)\log j \\ &=& (n+1)\sum_{j=1}^n \log j - \sum_{j=1}^n j\log j \\ &=& (n+1)\log n! - n^2 \frac{1}{n}\sum_{j=1}^n \frac{j}{n}\log \frac{j}{n} - \sum_{j=1}^n j\log n \\ &=& (n+1)\log n! - n^2\int_0^1 dx\ x\log x - \frac{n(n+1)}{2}\log n + O(n\log n) \\ &=& (n+1)\log n! +\frac{n^2}{4} - \frac{n(n+1)}{2}\log n + O(n\log n). \end{eqnarray*}$$ (The error estimate above can probably be tightened.)

Using Stirling's approximation we find $$\begin{eqnarray*} \frac{1}{n^2}\sum_{k=0}^n \log {n\choose k} &=& -\frac{n+1}{n^2}(\log n! - n\log n) - \frac{1}{2} + O\left(\frac{\log n}{n}\right) \\ &=& -\frac{n+1}{n^2}(-n + O(\log n)) - \frac{1}{2} + O\left(\frac{\log n}{n}\right) \\ &=& \frac{1}{2} + O\left(\frac{\log n}{n}\right). \end{eqnarray*}$$

The sum does not diverge, but converges. I suspect it converges to $1/2$ based on the value at $n=5000$ [link], but I've only managed to bound it by $3/4$.

From Wikipedia, $$\binom{n}{k} \leq \left(\frac{n \cdot e}{k}\right)^k$$

Then, \begin{align} x_n=\frac{1}{n^2}\sum_{k=0}^n\ln \binom{n}{k}&\leq \frac{1}{n^2}\sum_{k=0}^n\ln \left(\frac{n \cdot e}{k}\right)^k\\ &=-\frac{1}{n^2}\sum_{k=0}^n k\ln \left(\frac{k}{n \cdot e}\right)\\ &=-\frac{1}{n}\sum_{k=0}^n \left(\frac{k}{n}\right)\ln \left(\frac{k}{n \cdot e}\right)\end{align}

Looking at this as a Riemann Sum,

\begin{align} \lim_{n \to \infty} x_n &\leq \lim_{n \to \infty}-\frac{1}{n}\sum_{k=0}^n \left(\frac{k}{n}\right)\ln \left(\frac{k}{n \cdot e}\right) \\ &= \int_0^1 -x\ln\left(\frac{x}{e}\right)\mathrm{d}x \\ &=\left.\frac{x^2}{4}-\frac{x^2}{2}\ln x+\frac{x^2}{2}\ln e\right|_0^1 \\ &=\frac{3}{4} \end{align}

You can also bound it from below by $\frac{1}{4}$ by taking $\binom{n}{k} \geq \left(\frac{n}{k}\right)^k$.

I've managed to reduce it somewhat. $$x_n=\frac{1}{n^2}\ln(\prod_{k=0}^n \binom{n}{k})$$

$$x_n=\frac{1}{n^2}\ln \left ( \frac{n!}{(n-0)!0!}\frac{n!}{(n-1)!1!}\frac{n!}{(n-2)!2!}...\frac{n!}{(n-n)!n!}\right )$$

$$x_n=\frac{1}{n^2}\ln \left ( \frac{n!^{(n+1)}}{(0!1!2!...n!)^2} \right )$$

$$x_n=\frac{n+1}{n^2}\ln \left ( n! \right )-\frac{2}{n^2}\ln \left ( 0!1!2!...n! \right )$$

$$x_n=\frac{n+1}{n^2}\ln \left ( n! \right )-\frac{2}{n^2}\sum_{k=0}^n \ln(k!)$$

Using Stirling's approximation for large $n$ ($\ln(n!)=n \ln(n)-n$):

$$x_n=\frac{n+1}{n^2}(n \ln(n)-n)-\frac{2}{n^2}\sum_{k=0}^n \ln(k!)$$

$$x_n=\frac{n \ln(n)-n}{n}+\frac{n \ln(n)-n}{n^2}-\frac{2}{n^2}\sum_{k=0}^n \ln(k!)$$

$$x_n=\ln(n)-1+\frac{ \ln(n)-1}{n}-\frac{2}{n^2}\sum_{k=0}^n \ln(k!)$$

Noting that $\lim_{n \rightarrow \infty} \frac{\ln(n)}{n}=0$, for $n \rightarrow \infty$:

$$x_n=\ln(n)-1-\frac{2}{n^2}\sum_{k=0}^n \ln(k!)$$

$x_n=\frac{1}{n^2}\sum_{k=0}^{n}\ln{n\choose k}=\frac{1}{n^2}\ln(\prod {n\choose k})=\frac{1}{n^2}\ln\left(\frac{n!^n}{n!^2.(n-1)!^2(n-2)!^2...0!^2}\right)$ since ${n\choose k}=\frac{n!}{k!(n-k)!}$

$e^{n^2x_n}=\left(\frac{n^n(n-1)!}{n!^2}\right)e^{(n-1)^2x_{n-1}}=\left(\frac{n^{n-1}}{n!}\right)e^{(n-1)^2x_{n-1}}$

By Stirling's approximation, $n! \sim n^ne^{-n}\sqrt{2\pi n}$

$e^{n^2x_n}\sim \left(\frac{e^n}{n\sqrt{2\pi n}}\right)e^{(n-1)^2x_{n-1}}$

$x_n \sim \frac{(n-1)^2}{n^2}x_{n-1}+\frac{1}{n}-\frac{1}{n^2}\ln(n\sqrt{2\pi n})$

The $\frac{1}{n}$ term forces $x_n$ to tend to infinity, because $\frac{1}{n^2}\ln(n\sqrt{2\pi n})$ does not grow fast enough to stop it diverging.

$$\frac{1}{n^2}\sum_{k=0}^{n}ln\binom n k =\frac{1}{n^2}\sum_{k=0}^{n}ln\frac{n!}{k!(n-k)!} =\frac{1}{n^2}\sum_{k=0}^{n}(ln(n!)-ln(k!)-ln((n-k)!)) =\frac{1}{n^2}\sum_{k=0}^{n}ln(n!)-\frac{2}{n^2}\sum_{k=1}^{n}ln(k!) =\frac{n+1}{n^2}ln(n!)-\frac{2}{n^2}\sum_{k=0}^{n}ln(k!)$$

because ln(n!)~nln(n)-n,so

$$\frac{1}{n^2}\sum_{k=0}^{n}ln\binom n k =\frac{n+1}{n^2}(nln(n)-n)-\frac{2}{n^2}\sum_{k=1}^{n}(kln(k)-k)$$

then I am going to prove $\sum_{k=1}^{n}(kln(k)) \rightarrow \frac{n^2}{2}(ln(n)-1/2)$

$$\int_{i=x-1}^{x}(iln(i))dx<xln(x)<\int_{i=x}^{x+1}(iln(i))dx$$

and

$$\int(xln(x))dx=x^2/2(lnx-1/2)$$

so sum it up, we got $$\sum_{k=1}^{n}(kln(k)) \rightarrow \frac{n^2}{2}(ln(n)-\frac{1}{2})$$ then

$$\frac{1}{n^2}\sum_{k=0}^{n}ln\binom n k =\frac{n+1}{n^2}(nln(n)-n)-\frac{2}{n^2}\sum_{k=1}^{n}(kln(k)-k) =\frac{n+1}{n}(ln(n)-1)-\frac{2}{n^2}(\frac{n^2}{2}(lnn-\frac{1}{2})-\frac{n(n+1)}{2}) =ln(n)-1-(ln(n)-1/2-1)=1/2$$

For a two terms estimate of the behaviour of $\displaystyle\prod_{k=0}^{n}\binom{n}{k}$ one can check out March 2013 volume of Asymmetry problem V2-4 here (http://akotronismaths.blogspot.gr/p/asymmetry-electronic-mathematical.html)