求和Sn=1/(1*4)+1/(2*7)+.+1/n*(3n+1)

题目

求和Sn=1/(1*4)+1/(2*7)+.+1/n*(3n+1)

答案

1/n(3n+1)=3[(3n+1-3n)/3n(3n+1)]=3[1/3n-1/(3n+1)]Sn=3[(1/3+1/6+1/9+...+1/3n)-(1/4+1/7+1/10+...1/(3n+1))]定义ψ(k)=lim[n→∞](lnn-(1/k+1/(k+1)+...+1/n))则Sn=ψ(k+1)-ψ(k+4/3)+3-π√3/6-3ln3/2