Let $r=\frac{1}{2}$ , consider ${nr}^n$ for increasing value of $n$ and then find its value as $n$ tends to infinity.

If ${\left(20\right)}^{19}+2\left(21\right){\left(20\right)}^{18}+3{\left(21\right)}^2{\left(20\right)}^{17}+...+20{\left(21\right)}^{19}=k{\left(20\right)}^{19}$, then $k$ is equal to _____.

For $k\in \mathrm{N}$, if the sum of the series $1+\frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+......$ is $10$, then the value of $k$ is

If the value of ${\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\dots .. \mathrm{upto} \infty \right)}^{{\log }_{\left(0.25\right)}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots . \mathrm{upto} \infty \right)}$ is $l$, then $l^2$ is equal to             .