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Question Number 130886 by Khalmohmmad last updated on 30/Jan/21

Σ_(n=1) ^∞ ((1+2^n )/3^n )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}+\mathrm{2}^{{n}} }{\mathrm{3}^{{n}} } \\ $$

Answered by EDWIN88 last updated on 30/Jan/21

 Σ_(n=1) ^∞ ((1/3))^n +Σ_(n=1) ^∞ ((2/3))^n = ((1/3)/(1−(1/3)))+((2/3)/(1−(2/3)))   = (1/2)+(2/1)=(5/2)

$$\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{{n}} +\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{n}} =\:\frac{\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}}+\frac{\frac{\mathrm{2}}{\mathrm{3}}}{\mathrm{1}−\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{2}}{\mathrm{1}}=\frac{\mathrm{5}}{\mathrm{2}} \\ $$

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