Question

Find the sum of the following series $(1+2)lo{g}_{e}2+\frac{1+2^2}{\lfloor 2}(lo{g}_{e}2{)}^2+\frac{1+2^3}{\lfloor 3}(lo{g}_{e}2{)}^3+....$

Answer 1

$\Rightarrow e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

$e^{{\log }_{e}x}=1+{\log }_{e}x+\frac{{\left({\log }_{e}x\right)}^2}{2!}+\frac{{\left({\log }_{e}x\right)}^3}{3!}+....$

$e^{2{\log }_{e}x}=1+2{\log }_{e}x+\frac{2^2{\left({\log }_{e}x\right)}^3}{2!}+\frac{2^3{\left({\log }_{e}x\right)}^3}{3!}+....$

$e^{{\log }_{e}2}-1={\log }_{e}2+\frac{{\left({\log }_{e}2\right)}^2}{2!}+\frac{{\left({\log }_{e}2\right)}^3}{3!}+...$

$e^{2{\log }_{e}2}-1=2{\log }_{e}2+\frac{2^2{\left({\log }_{e}2\right)}^2}{2!}+\frac{2^2{\left({\log }_{e}2\right)}^3}{3!}+....$

$S=e^{{\log }_{e}2}-1+e^{2{\log }_{e}2}-1$

$=2+4-2$

$=4$