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

Kind regards

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Consider the function $$f(x)=(1+\frac{1}{x})^{x}$$.

Now try and show that $f'(x)>0$ for all $x>0$

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  • $\begingroup$ That's exactly what I want to prove. I can't use differentiation however. I tried proving it by induction, but I fail to prove this inequality $\endgroup$ – ParabolicAlcoholic Jun 6 '19 at 9:44
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Hint: Prove that $f(x)=(1+\frac{1}{x})^x$ is strictly increasing for x>0

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  • $\begingroup$ That's exactly what I want to prove. I can't use differentiation however. I tried proving it by induction, but I fail to prove this inequality $\endgroup$ – ParabolicAlcoholic Jun 6 '19 at 9:44
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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

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