Question Number 30213 by abdo imad last updated on 18/Feb/18
![p integr and p≥2 1) prove that ∃c∈ ]0,1[ / ln(ln(p+1))−ln(lnp) =(1/((p+c)ln(p+c))) 2)prove that ln(ln(p+1))−ln(ln(p))<(1/(plnp)) 3) prove that lim_(n→∞) Σ_(k=2) ^n (1/(klnk))=+∞ .](https://www.tinkutara.com/question/Q30213.png)
$${p}\:{integr}\:{and}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:\exists{c}\in\:\right]\mathrm{0},\mathrm{1}\left[\:/\right. \\ $$$${ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({lnp}\right)\:=\frac{\mathrm{1}}{\left({p}+{c}\right){ln}\left({p}+{c}\right)} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({ln}\left({p}\right)\right)<\frac{\mathrm{1}}{{plnp}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\mathrm{1}}{{klnk}}=+\infty\:. \\ $$