This question is motivated by a step in the proof given here.
$\begin{align*}
8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\
&=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\
&\geq (n+1)^8\left(8\left(\frac{9}{10}\right)^8\right)\\
&\gt (n+1)^8 .
\end{align*}$
I had no trouble following along with the proof until I hit the step that relied on
$$8\left(\frac{9}{10}\right)^8 > 1$$. So I whipped out a calculator and confirmed that this is indeed correct. And I could see, after some fooling around with pen and paper that any function in the form
\begin{align}
k \; \left(\frac{n}{n+1}\right)^k
\end{align}
where $n \in \mathbb{Z}$ and $k \rightarrow \infty$ is bound to fall below one and stay there. So it's not a given that any function in the above form will be greater than one.
What I'm actually curious about is whether there are nifty or simple little tricks or calculations you can do in your head or any handwavy type arguments that you can make to confirm that $$8\left(\frac{9}{10}\right)^8 > 1$$ and even more generally, to confirm for certain natural numbers $k,n$ whether
\begin{align}
k \; \left(\frac{n}{n+1}\right)^k > 1
\end{align}
So are there? And if there are, what are they?
It can be geometrical. It can use arguments based on loose bounds of certain log values. It doesn't even have to be particularly simple as long as it is something you can do in your head and it is something you can explain reasonably clearly so that others can do also it (so if you're able to mentally calculate numbers like Euler, it's not useful for me).
You can safely assume that I have difficulties multiplying anything greater two single digit integers in my head. But I do also know that $$\limsup_{k\rightarrow \infty} \log(k) – a\cdot k < 0$$ for any $a>0$ without having to go back to a textbook.
Best Answer
For this particular example, it's pretty easy, since $$ (9/10)^8 \geq (8/10)^4 \geq (6/10)^2 \geq (3/10) $$ and so you're done. Notice, my bounds are quite crude in the last two cases.
Repetitive squaring is the quickest way to get up to a large power. In fact, that's how a typical computer implementation will do it for integer powers.