The $n^{th}$ root of the geometric mean of binomial coefficients. [closed]

$\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean.

Prove $$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$

Solution 1:

$G_n$ is the geometric mean of $n+1$ numbers: $$ G_n=\left[\prod_{k=0}^n{n\choose k}\right]^{\frac1{n+1}} $$ or with $\log$ representing the natural logarithm (to the base $e$), $$ \log G_n = \frac1{n+1} \sum_{k=0}^n \log {n\choose k} = \log n! - \frac2{n+1} \sum_{k=0}^n \log k! \,. $$ Stirling's approximation is $n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ or $$ \log n! \approx \frac12\log{(2\pi n)}+n\log\left(\frac{n}{e}\right) = \left(n+\frac12\right)\log n+\frac12\log 2\pi-n $$ so $$ \eqalign{ \log \left(G_n\right)^\frac1n & = \frac1n \log G_n = \frac1n \log n! - \frac2{n(n+1)} \log \prod_{k=0}^n k! \\ & = \frac1n \log n! - \frac2{n(n+1)} \sum_{k=0}^n \log k! \\ & \approx \left(1+\frac1{2n}\right) \log n - \frac2{n(n+1)} \sum_{k=1}^n \left(k+\frac12\right)\log k - \frac1{2n}\log 2\pi \\ & \approx \left(1+\frac1{2n}\right) \log n - \frac2{n(n+1)} \left[ \frac{n(n+1)}{2}\log n - \frac{n(n+2)}{4} \right] - \frac1{2n}\log 2\pi \\ & = \frac{\log n-\log 2\pi}{2n} + \frac{n+2}{2(n+1)} \\ & \rightarrow \frac12 \,, } $$ where the sum of logarithms was approximated using the definite integrals $$ \sum_{k=1}^n \log k \approx \int_1^n \log x\,dx = \Big[x\log x-x\Big]_1^n \approx \Big[x\log x-x\Big]_0^n $$ and $$ \sum_{k=1}^n k \log k \approx \int_0^n x\log x\,dx=\left[\frac{x^2}{2}\log x - \frac{x^2}{4}\right]_0^n $$ (using integration by parts as shown in a comment), so that $$ \eqalign{ \sum_{k=1}^n \left(k+\frac12\right)\log k &= \sum_{k=1}^n k \log k + \frac12 \sum_{k=1}^n \log k \\ &\approx \left( \frac{n^2}{2}\log n - \frac{n^2}{4} \right) + \frac12 \Big( n \log n - n \Big) \\ &= \frac{n^2+n}{2}\log n - \frac{n^2+2n}{4} \,. } $$ Thus $$ G_n=e^{\log G_n}\rightarrow e^{\frac12}=\sqrt{e} \,. $$

Solution 2:

In fact, we have

$$ \lim_{n\to\infty}\left[\prod_{k=0}^{n}\binom{n}{k}\right]^{1/n^2} = \exp\left(1+2\int_{0}^{1}x\log x\; dx\right) = \sqrt{e}.$$

This follows from the identity

$$\frac{1}{n^2}\log \left[\prod_{k=0}^{n}\binom{n}{k}\right] = 2\sum_{j=1}^{n}\frac{j}{n}\log\left(\frac{j}{n}\right)\frac{1}{n} + \left(1+\frac{1}{n}\right)\log n - \left(1+\frac{2}{n}\right)\frac{1}{n}\log (n!),$$

together with the Stirling's formula.

In fact, I tried to write down the detailed derivation of this identity, but soon gave up since it's painstrikingly demanding to type $\LaTeX$ formulas in iPad2!

But you may begin with the identity

$$\log\binom{n}{k} = \log n! - \log k! - \log (n-k)!$$

and

$$ \log k! = \sum_{j=1}^{k} \log j,$$

and then you can change the order of summation.

Solution 3:

Hint

G is geometric mean:

$$G=\sqrt[n]{C_n^0C_n^1C_n^2\cdots C_n^n}$$

Hence,

$$\ln G=\frac{1}{n}\sum_{k=0}^n \ln C_k^n$$