frac{1}{2} + frac{1}{4} + frac{1}{8 times 2!} | vedclass

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(C) The given series is $S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8 	imes 2!} + \frac{1}{16 	imes 3!} + \frac{1}{32 	imes 4!} + \dots \infty$. We c

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$\frac{\sqrt{e}}{2}$

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(C) The given series is $S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8 	imes 2!} + \frac{1}{16 	imes 3!} + \frac{1}{32 	imes 4!} + \dots \infty$. We can rewrite this as $S = \frac{1}{2} \left[ 1 + \frac{1}{2} + \frac{1}{2^2 	imes 2!} + \frac{1}{2^3 	imes 3!} + \dots \infty ight]$. This is equivalent to $S = \frac{1}{2} \left[ 1 + \frac{(1/2)^1}{1!} + \frac{(1/2)^2}{2!} + \frac{(1/2)^3}{3!} + \dots \infty ight]$. Using the exponential series formula $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \infty$,we substitute $x = \frac{1}{2}$. Thus,$S = \frac{1}{2} e^{1/2} = \frac{\sqrt{e}}{2}$.

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8 \times 2!} + \frac{1}{16 \times 3!} + \frac{1}{32 \times 4!} + \dots \infty = $

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