PEEWSRIGWETRYqx

2022-01-21

How can I prove that $x-\frac{{x}^{2}}{2}<\mathrm{ln}\left(1+x\right)$

Linda Birchfield

Expert

Hint:
Prove that $\mathrm{ln}\left(1+x\right)-x+\frac{{x}^{2}}{2}$ is strictly increasing for $x>0$
edit: to see why this isn't a complete proof, consider ${x}^{2}-1$ for $x>0$. It's strictly increasing; does that show that ${x}^{2}>1$? I hope not, because it's not true!

Maria Lopez

Expert

consider $f\left(x\right)=\mathrm{ln}\left(1+x\right)-x+\frac{{x}^{2}}{2}$

Hence $f\left(x\right)>f\left(0\right)$

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