_ n 1 2 + 1 2^2 + 1 2^3 + ... + 1 2^n equals - Vidyadip

Question

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

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Answer

c
(c) $y = \mathop {\lim }\limits_{n \to \infty } \,\frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ....... + \frac{1}{{{2^n}}} = \mathop {\lim }\limits_{n \to \infty } \,\,\frac{1}{2}\,\frac{{\left[ {1 - {{\left( {\frac{1}{2}} \right)}^n}} \right]}}{{\left( {1 - \frac{1}{2}} \right)}}$

$\mathop {\lim }\limits_{n \to \infty } \,\left[ {1 - \frac{1}{{{2^n}}}} \right] = 1 - 0 = 1$

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