Solutions to Problems 966-974

Q.966 Prove that
$$\left(n\atop1\right)-{1\over2}\left(n\atop 2\right)
  +{1\over3}\left(n\atop 3\right)-\cdots\pm{1\over n}\left(n\atop n\right)
  =1+{1\over2}+{1\over3}+\cdots+{1\over n}\ ,$$
where the final sign on the left hand side is $+$ if $n$ is odd and $-$ if $n$ is even.