求证:1/(1*2)+1/(3*4)+...+1/((2n-1)*2n)=1/(n+1)+1/(n+2)+...+1/2n请给出证明并且解释.
问题描述:
求证:1/(1*2)+1/(3*4)+...+1/((2n-1)*2n)=1/(n+1)+1/(n+2)+...+1/2n
请给出证明并且解释.
答
1/(1*2)+1/(3*4)+...+1/((2n-1)*2n) =(1-1/2)+(1/3-1/4)+(1/5-1/6)+...+[1/(2n-1)-1/(2n)] =[1+1/3+1/5+..+1/(2n-1)]-[1/2+1/4+1/6+..+1/(2n)] =[1+1/3+1/5+..+1/(2n-1)]-1/2((1+1/2+1/3+..+1/n) --(1) 而 [1+1/...