I want to prove that $\left(1+\frac{1}{n}\right)^x>1+\frac{x}{n+1}$ $\forall x \ge1$ and $n \in \mathbb{N}$.
I first expanded $\left(1+\frac{1}{n}\right)^x=1+\frac{x}{n}+\frac{x(x-1)}{2n^2}+...$. So I wanted to truncate the expansion but I think the terms start becoming negative later on in the series. So I do not know how to do it. If anyone can give any hints it would be great. Thanks.