Tính tổng

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

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

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

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

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