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How would one prove the following:$$\left(1+\frac 1 n\right)^n < \left(1+\frac 1 {n+1}\right)^{n+1}$$

This is taken from the book challenge and thrill of precollege mathematics.

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    $\begingroup$ Is $n\in\Bbb N$? Also, what have you tried? $\endgroup$ – Dave Jun 29 '17 at 11:47
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One sledgehammer approach is to use calculus to verify that $x \ln(1+1/x)$ is monotonically increasing with $x>0.$

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  • $\begingroup$ Perhaps this problem is intended as a pre-log exercise? $\endgroup$ – Bernard Jun 29 '17 at 11:57
  • $\begingroup$ Probably, that's why I called it a sledgehammer approach. There are plenty of nice elementary solutions available at the link given by dromastyx. $\endgroup$ – Reiner Martin Jun 29 '17 at 12:04
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This is equivalent to showing that $(1+\frac{1}{n})^n$ is monotonically increasing. Here are many proofs:

I have to show $(1+\frac1n)^n$ is monotonically increasing sequence

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  • $\begingroup$ A comment would have sufficed. $\endgroup$ – StubbornAtom Jun 29 '17 at 12:02

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