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(C) The given expression is a geometric series with first term $a = \frac{1}{2}$ and common ratio $r = \frac{1}{2}$.The sum of the first $n$ terms of

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$1$

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(C) The given expression is a geometric series with first term $a = \frac{1}{2}$ and common ratio $r = \frac{1}{2}$.The sum of the first $n$ terms of a geometric series is given by $S_n = \frac{a(1 - r^n)}{1 - r}$.Substituting the values,we get $S_n = \frac{\frac{1}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} = \frac{\frac{1}{2}(1 - \frac{1}{2^n})}{\frac{1}{2}} = 1 - \frac{1}{2^n}$.Now,taking the limit as $n 	o \infty$:$\mathop {\lim }\limits_{n 	o \infty } (1 - \frac{1}{2^n}) = 1 - 0 = 1$.

$\mathop {\lim }\limits_{n \to \infty } \left( \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^n}}} \right)$ equals

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