Question

Find the value of $\frac{1}{lo{g}_{2}N}+\frac{1}{lo{g}_{3}N}+....+\frac{1}{{\log }_{2007}N}$, where $N=\mathrm{1.2.3........2007}$

Answer 1

$\frac{1}{{\log }_{2}N}+\frac{1}{{\log }_{3}N}+\frac{1}{{\log }_{3}N}.........\frac{1}{{\log }_{2007}N}=\frac{\log 2}{\log N}+\frac{\log 3}{\log N}+\frac{\log 4}{\log N}.........\frac{\log 2007}{\log N}=\frac{\log \left(\mathrm{2.3.4........2007}\right)}{\log N}=\frac{\log (2007!)}{\log (2007!)}=1$