Question Number 14152 by tawa tawa last updated on 28/May/17
Answered by ajfour last updated on 29/May/17
$${L}=\underset{\theta\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\:\left[\frac{\mathrm{tan}\:^{\mathrm{2}} \theta+\mathrm{tan}\:^{\mathrm{4}} \theta+\mathrm{tan}\:^{\mathrm{6}} \theta+…}{\left(\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \theta−\frac{\mathrm{5}}{\mathrm{6}}\right)+\left(\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \theta−\frac{\mathrm{5}}{\mathrm{6}}\right)^{\mathrm{2}} +…}\right] \\ $$$$\:\:\:\:\:{L}=\frac{\frac{\mathrm{1}}{\mathrm{3}}+\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} +…}{\frac{\mathrm{1}}{\mathrm{2}}+\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{3}} +…} \\ $$$$\:\:\:\:\:\:{L}=\frac{\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{1}−\mathrm{1}/\mathrm{3}}\right)}{\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{1}−\mathrm{1}/\mathrm{2}}\right)}=\frac{\frac{\mathrm{1}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{2}}}{\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{2}}{\mathrm{1}}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$$$ \\ $$
Commented by tawa tawa last updated on 29/May/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}. \\ $$