First slide

Special Series in Sequences and Series

Question

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

Moderate

A

2
B

3
C

4
D

5

Solution

$Let S = 1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots . (1)$
$\begin{array}{l} \frac{1}{3} S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{6}{3^{3}} + \frac{10}{3^{4}} + \dots . . . (2) \\ Subtracting (2) from (1) S (1 - \frac{1}{3}) = 1 + \frac{1}{3} + \frac{4}{3^{2}} + \frac{4}{3^{3}} + \frac{4}{3^{4}} + \dots . . \\ S (\frac{2}{3}) = \frac{4}{3} + \frac{4}{3^{2}} (1 + \frac{1}{3} + \frac{1}{3^{2}} + \dots . .) \\ \frac{2}{3} S = \frac{4}{3} + \frac{4}{3^{2}} (\frac{1}{1 - \frac{1}{3}}) = \frac{4}{3} + \frac{4}{3^{2}}, \frac{3}{2} = 2 ∴ S = 3 \end{array}$

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