Chào bạn!Mình giúp bạn 2 bài này nhé!
Bài 1:
[tex] S=1-2/2 C_n^1+2^2/3C_n^2-2^3/4C_n^3+...+(-1)^n.2^n/(n+1).C_n^n[/tex]
Ta có:
[TEX](1+x)^{n}=C_{n}^{0}+C_{n}^{1}x+C_{n}^{2}x^{2}+...+C_{n}^{n}x^{n}[/TEX]
[TEX]\Rightarrow \int_{-2}^{0}(1+x)^{n}dx=C_{n}^{0}\int_{-2}^{0}dx+C_{n}^{1}\int_{-2}^{0}x.dx+...+C_{n}^{n}\int_{-2}^{0}x^{n}.dx[/TEX]
[TEX]\Leftrightarrow \frac{(1+x)^{n+1}}{n+1} \mid^{0}_{-2}\ =x.C_{n}^{0}\mid_{-2}^{0}+\frac{x^{2}}{2}.C_{n}^{1} \mid^{0}_{-2}\ +...+ \frac{x^{n+1}}{n+1} \mid^{0}_{-2}\[/TEX]
[TEX]\Leftrightarrow \frac{1-(-2)^{n+1}}{n+1}=2.C_{n}^{0}-\frac{2^{2}}{2}.C_{n}^{1}+\frac{2^3}{3}.C_{n}^{2}-...+\frac{(-1)^{n+1}.2^{n+1}}{n+1}C_{n}^{n}[/TEX]
[TEX]\Leftrightarrow S=\frac{1-(-1)^{n+1}.2^{n+1}}{2(n+1)}[/TEX]