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For the series \( 2+\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)+\left((2 \sqrt{2}-1)+\frac{1}{2}\right)+\left((3 \sqrt{2}-2)+\frac{1}{2 \sqrt{2}}\right)+\ldots . \).
\( \mathrm{P} \)
(A) \( S_{n}=\sqrt{2}(\sqrt{2}+n-1)-n+\left(\frac{\left(2^{n / 2}-1\right)}{(\sqrt{2}-1) 2^{\frac{n-1}{2}}}\right) \)
(B) \( T_{n}=\sqrt{2}(\sqrt{2}+n-1)-n+\left(\frac{1}{2}\right)^{\frac{n-1}{2}} \)
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(C) \( S_{n}=\frac{n}{2}(3+(n-1) \sqrt{2}-n)+\left(\frac{\left(2^{n / 2}-1\right)}{(\sqrt{2}-1) 2^{\frac{n-1}{2}}}\right) \)
(D) \( S_{n}=\frac{n}{2}(3+(n-1) \sqrt{2}-n) \)
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