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时,在作出归纳假设后,需要证n=k+1时命题成立,即证

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时,在作出归纳假设后,
需要证n=k+1时命题成立,即证___________________

Answer 1

1-1/2+1/3-1/4...+1/2n-1-1/2n=1/n+1+1/n+2+...1/2n
当n=k时,等式
1-1/2+1/3-1/4...+1/(2k-1)-1/(2k)=1/(k+1)+1/(k+2)+...1/(2k)成立
那么当n=k+1时,需证明
1-1/2+1/3-1/4...+1/(2k-1)-1/(2k)+1/(2k+1)-1/(2k+2)
=1/(k+2)+1/(k+3)+.+1/(2k)+1/(2k+1)+1/(2k+2)