Chủ đề này có 1 trả lời

[Gioi han]

Chứng minh $1. C^1_n -\frac{1}{2} C^2_n ... +(-1)^{n-1}. C^n_n. \frac{1}{n}=1+ \frac{1}{2} ... +\frac{1}{n}$

[Mrnhan]

Chứng minh $1. C^1_n -\frac{1}{2} C^2_n ... +(-1)^{n-1}. C^n_n. \frac{1}{n}=1+ \frac{1}{2} ... +\frac{1}{n}$

 

LG.

 

Xét khai triển

 

$$\left ( 1+t \right )^n=\sum_{i=0}^{n}C_n^i t^n=1+C_n^1t+\cdots+C_n^nt^n$$

 

$$\Rightarrow C_n^1+C_n^2t+\cdots+C_n^nt^{n-1}=\frac{(1+t)^n-1}{t}=(1+t)^{n-1}+(1+t)^{n-2}+\cdots+1$$

 

$$\int_{-1}^{0}\left [ C_n^1+C_n^2t+\cdots+C_n^nt^{n-1} \right ]dt=\int_{-1}^{0}\left [ (1+t)^{n-1}+(1+t)^{n-2}+\cdots+1 \right ]dt $$

 

$$\Leftrightarrow C_1^n-\frac{1}{2}C_n^2+\cdots+ \frac{(-1)^{n-1}}{n}C_n^n=\frac{1}{n}+\frac{1}{n-1}+\cdots +1$$