# How to solve $\left(1+\frac{1}{n+2}\right)^{n+2}\geq \left(1+\frac{1}{n+1}\right)^{n+1}$ for all $n\in \mathbb{N}$ [duplicate]

any impulses, suggestions? I have been trying for a while but it doesn't get me anywhere...

Kind regards

• – Martin R Jun 6 '19 at 9:49
• Don't you have to prove it via Induction? – ParabolicAlcoholic Jun 6 '19 at 9:59

Consider the function $$f(x)=(1+\frac{1}{x})^{x}$$.
Now try and show that $$f'(x)>0$$ for all $$x>0$$
Hint: Prove that $$f(x)=(1+\frac{1}{x})^x$$ is strictly increasing for x>0
log both sides and rearrange:sufficient to prove $$\frac{n+2}{n+1}\ge\ln(\frac{\frac{n+2}{n+1}}{\frac{n+3}{n+2}})$$ which is $$\ln(e^{\frac{n+2}{n+1}})\ge\ln(1+\frac{1}{(n+1)(n+3)})$$ exponential both sides and LHS$$>2>$$RHS