Question Number 196094 by leicianocosta last updated on 18/Aug/23
Answered by dimentri last updated on 18/Aug/23
$$\left(\mathrm{1}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}^{\mathrm{3}\left({n}−\mathrm{1}\right)} }{\mathrm{3}^{\mathrm{2}{n}} }\right)=\:\frac{\mathrm{1}}{\mathrm{9}}+\frac{\mathrm{8}}{\mathrm{81}}+\frac{\mathrm{64}}{\mathrm{729}}+\ldots \\ $$$$\:\:\:\:\:\:=\:\frac{\frac{\mathrm{1}}{\mathrm{9}}}{\mathrm{1}−\frac{\mathrm{8}}{\mathrm{9}}}\:=\:\mathrm{1}\: \\ $$