数列\( 1\frac{1}{2},3\frac{1}{4},5\frac{1}{8},7\frac{1}{16},\cdots ,\)前n项和为 ( )
A. \( {n}^{2}-\frac{1}{{2}^{n}}+1\)
B. \( {n}^{2}-\frac{1}{{2}^{n+1}}+\frac{1}{2}\) C. \( {n}^{2}-n-\frac{1}{{2}^{n}}+1\) D. \( {n}^{2}-n\frac{1}{{2}^{n+1}}+\frac{1}{2}\)
