Prove that the sequence (1+1/n)^(n+1) is monotonically decreasing,without using functions and derivatives.

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- Dec 5th 2012, 08:59 AMhediprove monotonicity of a sequence
Prove that the sequence (1+1/n)^(n+1) is monotonically decreasing,without using functions and derivatives.

- Dec 5th 2012, 09:34 AMemakarovRe: prove monotonicity of a sequence
See here.

- Dec 5th 2012, 11:18 AMPlatoRe: prove monotonicity of a sequence

$\displaystyle \begin{align*}\left( {1 - \frac{1}{n}} \right)^n\left( {1 + \frac{1}{n}} \right)^n&= \left( {1 - \frac{1}{n^2}} \right)^n \ge\left( {1 - \frac{1}{n}} \right)\\ \left( {1 + \frac{1}{n}} \right)^n &\ge \left( {1 - \frac{1}{n}} \right)^{1-n}\\\left( {1 + \frac{1}{n}} \right)^n &\ge\left( {1 + \frac{1}{n-1}} \right)^{n-1} \end{align*}$. - Dec 5th 2012, 11:57 AMhediRe: prove monotonicity of a sequence
Thank's but this is not the right sequence.The given sequence is monotonically decreasing