Hey For A Sequence $\displaystyle a_n=n^{\frac{1}{n}}-1$ Where $\displaystyle a_n>0,n>1$ How Can I Deduce That $\displaystyle \forall n>1,n-1\geq\frac{1}{2}n\left(n-1\right)a_n^2$
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Rearrange as $\displaystyle n=(1+a_n)^n$ and binomial expansion $\displaystyle n=1+na_n+\frac{n(n-1)}2a_n^2+\dots>1+\frac{n(n-1)}2a_n^2$ does the trick.
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