# prove monotonicity of a sequence

• Dec 5th 2012, 08:59 AM
hedi
prove 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 AM
emakarov
Re: prove monotonicity of a sequence
See here.
• Dec 5th 2012, 11:18 AM
Plato
Re: prove monotonicity of a sequence
Quote:

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

\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 AM
hedi
Re: prove monotonicity of a sequence
Thank's but this is not the right sequence.The given sequence is monotonically decreasing