If $a_n=n\left(1-\frac{1}{n}\right)^{n[\log n]}$, prove $1\leqslant \liminf a_n$ and $\limsup a_n\leqslant e.$

Question

Prove that: $$\limsup n\left(1-\frac{1}{n}\right)^{n[\log n]} \leqslant e$$ and $$\liminf n\left(1-\frac{1}{n}\right)^{n[\log n]} \geqslant 1.$$

Attempt. Since for $n>2$: $$1<[\log n]<\log n+1\leqslant n,$$ we would get: $$n\left(1-\frac{1}{n}\right)^{n[\log n]}\leqslant n\left(1-\frac{1}{n}\right)^n \to +\infty,$$ $$n\left(1-\frac{1}{n}\right)^{n[\log n]}\geqslant n\left(1-\frac{1}{n}\right)^{n^2} \to 0,$$

so the above estimates are not convenient.

Thanks for the help.

Answer 1

First, note that $\log n-1 < [\log n] \leq \log n$. Now, $0<1- \frac{1}{n}<1$, so we have

$$ n \left( 1- \frac{1}{n} \right)^{n \log n} \leq n \left( 1- \frac{1}{n} \right)^{n [\log n]} < n \left( 1- \frac{1}{n} \right)^{n (\log n - 1)}. $$

We now want to use the fact that $\left(1- \frac{1}{n}\right)^{n} \to e^{-1}$ as $n \to \infty$ to simplify the above inequalities. To make the computation easier, write this as $n\log(1- \frac{1}{n}) \to -1$, which can be sharpened to $n\log(1- \frac{1}{n}) = -1 - \frac{1}{2n} + O\left( \frac{1}{n^2} \right)$ using the Taylor series for $\log(1-x).$

Now,

$$\log \left(n \left( 1- \frac{1}{n} \right)^{n \log n} \right) = \log n \left( 1 + n \log \left(1 - \frac{1}{n} \right) \right) = \log n\left(- \frac{1}{2n} + O\left( \frac{1}{n^2}\right) \right) \to 0$$

as $n \to \infty$, and similarly

$$ \log \left( n \left( 1- \frac{1}{n} \right)^{n (\log n - 1)} \right) = \log n \left( 1 + n \log \left(1 - \frac{1}{n} \right) \right) - n \log\left( 1- \frac{1}{n} \right) \to 1$$

as $n\to\infty.$ Taking the exponentials of both of these inequalities gives the desired result.

Answer 2

Actually, the correct inequalities for $n>1$ are:

$\log n-1<[\log n] < \log n$, (noting that $\log n$ is never an integer for $n>1$, or you can use less or equal in the right one as it doesn't really matter ) so you get $n\left(1-\frac{1}{n}\right)^{n[log n]} < n\left(1-\frac{1}{n}\right)^{(n\log n -n)}$ and the limit of RHS is obviously $e$, since $n\left(1-\frac{1}{n}\right)^{n\log n}$ converges to $1$ (by taking logarithms and using that $\log (1-\frac{1}{n}) = -\frac{1}{n} + O(\frac{1}{n^2})$, so multiplying by $n\log n$ you get just an $O(\frac{\log n}{n})$ which converges to $0$ hence the claim). The other limit is now obvious since $n\left(1-\frac{1}{n}\right)^{n[log n]} > n\left(1-\frac{1}{n}\right)^{n\log n}$ which converges to $1$ as noted above.