Proof that $(1+\epsilon)^n = O(1+n\epsilon)$

Question

How to prove that $(1+\epsilon)^n = O(1+n\epsilon)$ ?

So far I proved the following:

By the binomial, $(1+\epsilon)^n > 1+n\epsilon$

Also $\epsilon^n$ = 0 when n-> infinity.

Edit: n constant. $\epsilon$ -> 0.

Answer 1

$(1+\epsilon)^n=e^{n \times log(1+\epsilon)}$

$=1+n \times log(1+\epsilon)+\frac{(n \times log(1+\epsilon))^2}{2!}+...,$ Taylor series

$=1+n\epsilon+... ,$ Please develop this serie well to see what is happening, since $\epsilon^n\simeq0$ and $log(1+\epsilon)=\sum_{n=0}^{\infty}(-1)^n\frac{\epsilon^{n+1}}{n+1}$, the result then follow:

$(1+\epsilon)^n=\circ(1+n\epsilon)$