How to prove this is a decreasing sequence?

Question

I have this sequence: $\{a_n\} = {\frac {1}{n^{1+\frac {1}{n}}}}$ I tried to take the derivative to show the rate of change is negative for some n but the derivative is not very nice to work with. I tried to show $\frac {\{a_{n+1}\}}{\{a_n\}}<1$ but again the ratio didn't clearly show that to me. What other method can I try to clearly show as n increases the sequence decreases? EDIT: Also is there any property to say that if a sequence is bounded above by a decreasing sequence, then our sequence is also decreasing? Like $\frac {1}{n}$ in this case?

Answer 1

Being bounded above by a decreasing sequence does not imply that a sequence is decreasing.

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Since the logarithm is an increasing function, it is sufficient to show that

$$\ln n^{1 + \frac 1 n}$$

is an increasing function. But this just reduces to studying

$$\left(1 + \frac 1 n\right) \ln n.$$

The derivative of this is

$$\left(1 + \frac 1 n\right) \frac 1 n - \frac{\ln n}{n^2} \approx \frac 1 n > 0$$

once $n$ is sufficiently large.