Showing that an unusual oscillating sequence diverges.

Question

I am working on the following problem:

Let $s_n=(\frac{1}{n}-1)^n$. Show that $s_n$ diverges.

I have successfully shown that it cannot converge to any nonzero value. I am struggling, however, with showing that it cannot converge to $0$.

My main idea is to consider the "even" subsequence, which has all positive values. If I can show that the even subsequence is monotone increasing, then I can of course show that $0$ cannot be a limit. Unfortunately, I am stuck at the following step:

I need to show that $$ \left(1-\frac{1}{2n+2}\right)^{2n+2} > \left(1-\frac{1}{2n}\right)^{2n} $$

which I cannot prove. Can someone help me out with this one?

Answer 1

With some algebra, we get that: $$\left(\frac{1}{n}-1\right)^n = (-1)^n\left(1-\frac{1}{n}\right)^n.$$

Moreover, we know that $$\left(1-\frac{1}{n}\right)^n \to e^{-1}.$$

This means that it is bounded, but it does not converge. After a while (i.e. sufficiently long time, or sufficiently big $n$), it just oscillates between the values $-e^{-1}$ and $+e^{-1}$.