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Math Help - prove monotonicity of a sequence

  1. #1
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    prove monotonicity of a sequence

    Prove that the sequence (1+1/n)^(n+1) is monotonically decreasing,without using functions and derivatives.
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    Re: prove monotonicity of a sequence

    See here.
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    Re: prove monotonicity of a sequence

    Quote Originally Posted by hedi View Post
    Prove that the sequence (1+1/n)^(n+1) is monotonically decreasing,without using functions and derivatives.

    \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*}.
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    Re: prove monotonicity of a sequence

    Thank's but this is not the right sequence.The given sequence is monotonically decreasing
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