36+36⋅59+3⋅5⋅76⋅9⋅12+……∞=

S=36+36⋅59+3⋅5⋅76⋅9⋅12+……∞=⇒S=31.33.6+1.3.53.6.9+……⇒S=31+13+1.33.6+……−4⇒S=31+11!13+1.32!132+...-4⇒S=31-23-12-4⇒S=33-4

36+36⋅59+3⋅5⋅76⋅9⋅12+……∞=

S=36+36⋅59+3⋅5⋅76⋅9⋅12+……∞=⇒S=31.33.6+1.3.53.6.9+……⇒S=31+13+1.33.6+……−4⇒S=31+11!13+1.32!132+...-4⇒S=31-23-12-4⇒S=33-4

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$\frac{3}{6} + \frac{3}{6} \cdot \frac{5}{9} + \frac{3 \cdot 5 \cdot 7}{6 \cdot 9 \cdot 12} + \dots \dots \infty =$

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$2 \sqrt{3} - 4$

$3 \sqrt{3} - 2$

$3 \sqrt{3} - 4$

$2 \sqrt{3} + 4$

answer is C.

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Detailed Solution

$\begin{array}{l} S = \frac{3}{6} + \frac{3}{6} \cdot \frac{5}{9} + \frac{3 \cdot 5 \cdot 7}{6 \cdot 9 \cdot 12} + \dots \dots \infty = \\ \Rightarrow S = 3 (\frac{1 .3}{3 .6} + \frac{1 .3.5}{3 .6.9} + \dots \dots) \\ \Rightarrow S = 3 \{1 + \frac{1}{3} + \frac{1 .3}{3 .6} + \dots \dots\} - 4 \\ \Rightarrow S = 3 \{1 + \frac{1}{1!} (\frac{1}{3}) + \frac{1.3}{2!} {(\frac{1}{3})}^{2} + . . .\} - 4 \\ \Rightarrow S = 3 {(1 - \frac{2}{3})}^{\frac{- 1}{2}} - 4 \\ \Rightarrow S = 3 \sqrt{3} - 4 \end{array}$