A) \[n-1+{{2}^{-n}}\]
B) \[1\]
C) \[n-1\]
D) \[1+{{2}^{-n}}\]
E) \[n-1-{{2}^{-n}}\]
Correct Answer: A
Solution :
Let\[{{S}_{n}}\]be the sum of first n terms of the series \[\therefore \]\[{{S}_{n}}=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....\] \[\Rightarrow \]\[{{S}_{n}}=\left( 1-\frac{1}{2} \right)+\left( 1-\frac{1}{4} \right)+\left( 1-\frac{1}{8} \right)+....\] \[\Rightarrow \]\[{{S}_{n}}=n-\frac{1}{2}\frac{(1-1/{{2}^{n}})}{(1-1/2)}=n-1+\frac{1}{{{2}^{n}}}\] \[=n-1+{{2}^{-n}}\]
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