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Prove that:-

For any natural number n:- \[n \geq 2(n!)^{\frac{1}{n}}-1\]

Please post a non induction proof

Note by Aman Sharma 3 years, 1 month ago

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This can be re-arranged to this inequality

\({ \left( 1+\frac { 1 }{ n } \right) }^{ n }\ge 2\displaystyle\prod _{ k }^{ n }{ \frac { k }{ n } } \)

For \(n=1\), we have \(2=2\). Thereafter, the left side approaches \(e\) while the right side drops towards \(0\)

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Realy intresting proof

Take numbers 1,2,3.....n Then \(A.M \geq G.M\)

I also did in the same way

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TopNewestThis can be re-arranged to this inequality

\({ \left( 1+\frac { 1 }{ n } \right) }^{ n }\ge 2\displaystyle\prod _{ k }^{ n }{ \frac { k }{ n } } \)

For \(n=1\), we have \(2=2\). Thereafter, the left side approaches \(e\) while the right side drops towards \(0\)

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Realy intresting proof

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Take numbers 1,2,3.....n Then \(A.M \geq G.M\)

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I also did in the same way

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