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Proving $f(n)=\left(1+\frac{1}{n}\right)^{n+\frac{1}{2}}$ is a decreasing function, within the interval $(1,\infty)$.

Typically I would take the derivative of $f(n)$ and set it to 0 and find the critical points. Then I would construct a number, choose random numbers between each interval and I would be able to tell if it is increasing or decreasing within an interval. But $f(n)$ doesn't have a critical point.

How would I do this?

Username Unknown
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1 Answers1

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Note that its derivative, namely $$f'(n)=D\Big[ \left(1+\frac{1}{n}\right)^{n+\frac{1}{2}} \Big] = \left(\frac{1}{n}+1\right)^{n+\frac{1}{2}} \left(\ln \left(1 + \frac{1}{n}\right)-\frac{n+\frac{1}{2}}{\left(\frac{1}{n}+1\right) n^2}\right) < 0 \quad \forall n \in [1,\infty[$$ and this implies the function $f(n)=\left(1+\frac{1}{n}\right)^{n+\frac{1}{2}}$ is decreasing in $[1,\infty[$.

sirfoga
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