Question

用数学归纳法证明:1-1/2+1/3-1/4+...+1/2n-1-1/2n=1/n+1+1/n+2+...+1/2n

Answer 1

1,
n=1时，左边=1-1/2=1/2.右边=1/2成立
2，设n=k时成立就是
1-1/2+1/3-1/4+...+1/(2k-1)-1/2k=1/(k+1)+...1/(2k)
当
n=k+1时，则1-1/2+1/3+...+1/(2k-1)-1/2k+1/(2k+1)-1/(2k+2)=1/(k+1)+...1/(2k)+1/(2k+1)-1/(2k+2)=1/(k+2)+...+1/(2k)+1/(2k+1)+1/(k+1)-1/(2k+2)
下面证明
1/(k+1)-1/(2k+2)=1/(2k+2)???
1/(k+1)-1/(2k+2)=(2-1)/(2k+2)=1/(2k+2)
！！！
所以
1-1/2+1/3+...+1/(2k-1)-1/2k+1/(2k+1)-1/(2k+2)
=
1/(k+1)+...1/(2k)+
1/(2k+1)+1/（2k+2）就是说
n=k+1时成立所以对于一切n都会成立