How can I show the inequality $\ln(\ln(k+1))-\ln(\ln(k))<\dfrac{1}{k\ln(k)},\forall K\in \mathbb{N}, K\geq 2.$
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From Mean value theorem we have, $$\displaystyle\frac{\ln(\ln(k+1))-\ln(\ln(k))}{k+1-k}=\frac{1}{\ln c}.\frac{1}{c},c\in(k,k+1)$$ $$k<c\Rightarrow\frac{1}{\ln c}.\frac{1}{c}<\frac{1}{\ln k}.\frac{1}{k}$$ $$\Rightarrow\displaystyle{\ln(\ln(k+1))-\ln(\ln(k))}=\frac{1}{\ln c}.\frac{1}{c}<\frac{1}{\ln k}.\frac{1}{k}$$ $$\Rightarrow\displaystyle{\ln(\ln(k+1))-\ln(\ln(k))}<\frac{1}{\ln k}.\frac{1}{k}$$ We are done. |
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