The value of sum_{n=0}^{infty} frac{(n+1)^2}{ | vedclass

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Question

(A) Let $S = \sum_{n=0}^{\infty} \frac{(n+1)^2}{7^n} = \frac{1^2}{7^0} + \frac{2^2}{7^1} + \frac{3^2}{7^2} + \frac{4^2}{7^3} + \dots$Multiply by $\fra

Vedclass · Accepted Answer

$\frac{49}{27}$

Vedclass · Answer

(A) Let $S = \sum_{n=0}^{\infty} \frac{(n+1)^2}{7^n} = \frac{1^2}{7^0} + \frac{2^2}{7^1} + \frac{3^2}{7^2} + \frac{4^2}{7^3} + \dots$Multiply by $\frac{1}{7}$:$\frac{S}{7} = \frac{1^2}{7^1} + \frac{2^2}{7^2} + \frac{3^2}{7^3} + \dots$Subtracting the two equations:$S - \frac{S}{7} = 1 + \frac{2^2 - 1^2}{7^1} + \frac{3^2 - 2^2}{7^2} + \frac{4^2 - 3^2}{7^3} + \dots$$\frac{6S}{7} = 1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots$This is an arithmetico-geometric series. Let $T = 1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots$$\frac{T}{7} = \frac{1}{7} + \frac{3}{7^2} + \frac{5}{7^3} + \dots$Subtracting these:$T - \frac{T}{7} = 1 + \frac{2}{7} + \frac{2}{7^2} + \frac{2}{7^3} + \dots$$\frac{6T}{7} = 1 + \frac{2/7}{1 - 1/7} = 1 + \frac{2/7}{6/7} = 1 + \frac{1}{3} = \frac{4}{3}$$T = \frac{4}{3} 	imes \frac{7}{6} = \frac{14}{9}$Since $\frac{6S}{7} = T$,we have $\frac{6S}{7} = \frac{14}{9} \Rightarrow S = \frac{14}{9} 	imes \frac{7}{6} = \frac{98}{54} = \frac{49}{27}$.

The value of $\sum_{n=0}^{\infty} \frac{(n+1)^2}{7^n}$ is -

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