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My goal is to show that the sequence

$$\left(\frac{1}{n} +1\right)^n = e^1$$

is strictly increasing. My attempt was to show this by induction. After checking for $n=1$ and $n=2$ I assumed it to be true for $n-$ and $n$:

$$\left(\frac{1}{n-1} +1\right)^{n-1} < \left(\frac{1}{n} +1\right)^n = $$

And then to show the inequality for $n$ and $n+1$. This led me nowhere. Another attempt was to secretly assume that I can view $n$ as a real parameter and then differentiate with respect to $n$. My thought was that if the resulting derivative was strictly positive for positive $n$ or rather $n\geq 1$ then the statement would hold. But I strongly suspect that this is not so easy as it seems.

Can anyone offer help?

Jakob Elias
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1 Answers1

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You want to prove $$\left(\frac{n+1}{n}\right)^{n}>\left(\frac{n}{n-1}\right)^{n-1}$$ for $n\ge2$. This is $$(n+1)^n(n-1)^{n-1}>n^{2n+1}.$$ AM/GM gives $$(n+1)^n(n-1)^{n-1}\ge\left(n+\frac1{2n-1}\right)^{2n+1}.$$

Angina Seng
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