Tính tổng 

Câu hỏi: Tính tổng $S=1-\frac{1}{3}{C}_n^1+\frac{1}{5}{C}_n^2-\frac{1}{7}{C}_n^3+\mathrm{...}+\frac{{\left(-1\right)}^n}{2n+1}{C}_n^n$

A. $S=\frac{\mathrm{2.4.6...2}n}{\mathrm{3.5.7...}\left(2n+1\right)}$

B. $S=\frac{\left(2n\right)!}{\left(n+1\right)!}$

C. $S=\frac{{\left(-1\right)}^{n}n!\left(n+1\right)!}{\left(2n\right)!}$

D. $S=\frac{{\left(-1\right)}^{2n+1}\left(2n\right)!}{\left(2n+1\right)!}$