Giá trị của \(\mathrm{C}_{n}^{0}-\frac{1}{2} \mathrm{C}_{n}^{1}+\frac{1}{3} \mathrm{C}_{n}^{2}-\cdots+\frac{(-1)^{n}}{n+1} \mathrm{C}_{n}^{n}\) là:

\((1+x)^{n}=\mathrm{C}_{n}^{0}+\mathrm{C}_{n}^{1} x+\mathrm{C}_{n}^{2} x^{2}+\cdots+\mathrm{C}_{n}^{n} x^{n}\)

\(x=\frac{1}{3} \text { ta được }\)

\(\begin{aligned}
&\left(1+\frac{1}{3}\right)^{n}=\mathrm{C}_{n}^{0}+\frac{1}{3} \mathrm{C}_{n}^{1}+\frac{1}{3^{2}} \mathrm{C}_{n}^{2}+\cdots+\frac{1}{3^{k}} \mathrm{C}_{n}^{k}+\cdots+\frac{1}{3^{n}} \mathrm{C}_{n}^{n} \\
\Rightarrow & 3^{n}\left(1+\frac{1}{3}\right)^{n}=3^{n} \mathrm{C}_{n}^{0}+\frac{1}{3} \mathrm{C}_{n}^{1}+\frac{1}{3^{2}} \mathrm{C}_{n}^{2}+\cdots+\frac{1}{3^{k}} \mathrm{C}_{n}^{k}+\cdots+\frac{1}{3^{n}} \mathrm{C}_{n}^{n} \\
\Rightarrow & S_{1}=3^{n}\left(\frac{4}{3}\right)^{n}=4^{n}
\end{aligned}\)