A) \[n-1-{{2}^{-n}}\]
B) 1
C) \[n-1+{{2}^{-n}}\]
D) \[1+{{2}^{-n}}\]
Correct Answer: C
Solution :
\[\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+...\] \[=\left( 1-\frac{1}{2} \right)+\left( 1-\frac{1}{4} \right)+\left( 1-\frac{1}{8} \right)+\left( 1-\frac{1}{16} \right)+....\] \[=n-\frac{\frac{1}{2}\left\{ 1-\frac{1}{{{2}^{n}}} \right\}}{1-\frac{1}{2}}=n-1+{{2}^{-n}}\]
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