The sum to infinity of the series 1 + 2 3 + 6 3 2 + 10 3 3 + 14 3 4 + ? is

Question

The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ is

Smart Achievers · Accepted Answer

(1)

$Let S = 1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ ...(i)

$\frac{1}{3} S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{6}{3^{3}} + \frac{10}{3^{4}} + \dots$ ...(ii)

$Subtracting (ii) from (i), we get$

$\frac{2}{3} S = 1 + \frac{1}{3} + \frac{4}{3^{2}} + \frac{4}{3^{3}} + \frac{4}{3^{4}} + \dots$

$= \frac{4}{3} [1 + \frac{1}{3} + \dots to \infty] = \frac{4}{3} \cdot \frac{1}{1 - \frac{1}{3}} = \frac{4}{3} \cdot \frac{1}{\frac{2}{3}} = 2$

$\Rightarrow$ $\frac{2}{3} S = 2 \Rightarrow S = 3$

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