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For which values of $x\in\Bbb R$ is the sequence $a_n:=\left(1+\frac xn\right)^n$ monotone increasing? Thanks in advance!

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Hint: $$ 1 + \frac{x}{n+1} = \frac{1}{n+1}\left( 1 + \underbrace{(1+\frac{x}{n}) + .. + (1+\frac{x}{n})}_{\text{n terms}} ) \right) $$ –  achille hui Feb 9 '13 at 2:01
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1 Answer

What does it mean for a sequence to be monotone increasing?

Hint:

You need to find for which $x \in \mathbb{R}$ it is the case that $$a_n = \left(1 + \frac xn \right)^n \;\;\leq \;\;a_{n+1} = \left(1 + \frac{x}{n+1}\right)^{n+1}, \;\;\forall n\in \mathbb{N}$$

Hint 2: Note that $$\left(1 + \frac{x}{n+1}\right) = \frac{1}{n+1}\left( 1 + \underbrace{(1+\frac{x}{n}) + .. + (1+\frac{x}{n})}_{\text{n summands}}\right) $$

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Yes... Thank you. I am quite aware of what it means. I wanted an elegant approach. Without having to mess around with binomial coefficients. –  user61541 Feb 9 '13 at 0:47
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@user61541 That's why it helps, when you post a question, to provide context and clarify what you understand, and what you've tried, and what you think might work...so people don't tell you what you already know. –  amWhy Feb 9 '13 at 0:50
    
I think I had an answer depicting like you did here before. + –  B. S. Feb 9 '13 at 19:39
    
i dont understand yet! can u pleez define in any other way??? –  user97328 Sep 27 '13 at 18:09
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