Any elementary proof of the monotonicity of $a_{n} =(1+\frac{1}{n})^{n+\frac{1}{2}}$?

Question

We do have the elementary proof of the monotonicity of $(1+\frac{1}{n})^{n}$ and $(1+\frac{1}{n})^{n+1}$ .

Here is an example.

First we have $$\ln \left(1+\frac{1}{n+1}\right) = \int_{1}^{1+\frac{1}{n+1}}\frac{1}{x}\,dx>\frac{1}{n+2}=\frac{n\left(n+1\right)}{n+2}\int_{1+\frac{1}{n+1}}^{1+\frac{1}{n}}\,dx > n\int_{1+\frac{1}{n+1}}^{1+\frac{1}{n}}\frac{1}{x}\,dx$$ Now add $$n\ln \left(1+\frac{1}{n+1}\right) = n\int_{1}^{1+\frac{1}{n+1}}\frac{1}{x}\,dx$$ We get $$\left(n+1\right)\ln \left(1+\frac{1}{n+1}\right) > n\int_{1}^{1+\frac{1}{n}}\frac{1}{x}\,dx = n\ln \left(1+\frac{1}{n}\right)$$

So can we solve the monotonicity of $a_{n} =(1+\frac{1}{n})^{n+\frac{1}{2}}$ similiarly? Or by any other elegant elementary proof?

Answer 1

$$ \left(n+\tfrac{1}{2}\right)\log\left(1+\tfrac{1}{n}\right)=\int_{0}^{1}\frac{n+\tfrac{1}{2}}{x+n}\,dx= \int_{-1/2}^{1/2}\frac{1}{1+\frac{2x}{2n+1}}\,dx = \int_{0}^{1/2}\frac{2}{1-\left(\frac{2x}{2n+1}\right)^2}\,dx $$ produces

$$ \left(n+\tfrac{1}{2}\right)\log\left(1+\tfrac{1}{n}\right)= \int_{0}^{1}\frac{dx}{1-\left(\frac{x}{2n+1}\right)^2} $$

hence it is clear that the RHS is decreasing, since for any $x\in(0,1)$ and any $N>n$ we have $$ \frac{1}{1-\left(\frac{x}{2N+1}\right)^2} < \frac{1}{1-\left(\frac{x}{2n+1}\right)^2}.$$

Answer 2

The second derivative of $\log a_n$ is positive,
so the first derivative is increasing, with limit 0,
so the first derivative is negative,
so the sequence is monotonic decreasing.