I have to determine which number is greater, $2\sqrt{2}$ or $e$.
I had a similar question as well, it was $2^\sqrt{2}$ compared to $e$.
For that one I managed to prove the inequality by using the increasing sequence converging to $e$: $(1+\frac1n )^n $
So I just searched for a value to assign to n such that $(1+\frac1n )^n \gt 2^\sqrt2$
I tried to proceed in a similar way with $2\sqrt{2} \gt \lt e$ , but it seems I can't get nowhere (I used the sequence decreasing and converging to $e$: $(1+\frac1n )^{n+1}$ )
Is there another way to prove the inequality without the use of the calculator and maybe using derivatives? The question was in a derivatives file, so I'm wondering is there's a way to get to the end using them.
Any hint would be much appreciated, thanks.