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Why is the sequence $\frac{1}{\log(n)}$ a convex sequence?

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Well, we happen to know that the function $f(x) = \dfrac{1}{\log x}$ is convex for $x > 1$ as its second derivative, $\dfrac{\log x + 2}{x^2 \log ^3 x}$ is positive for $x > 1$.

So if you start your sequence at 2, all is good. As I don't believe you would start it at 1, as $\log(1)^{-1}$ is... difficult to define.