Question

怎么解答这个题:1+一加二分之一再加一加二加三分之一再依次往下加,最后是一加二加三加四一直加到100

Answer 1

1+1/(1+2)+1/(1+2+3)+...+1/(1+2+3+...100)=?

1/(1+2+3+...n)=1/[n(1+n)/2]=2/n(n+1)=2(1/n-1/(n+1))

所以原式=2(1-1/2)+2(1/2-1/3)+2(1/3-1/4)+...+2(1/100-1/101)

=2(1-1/2+1/2-1/3+1/3-1/4+...+1/100-1/101)

=2(1-1/101)

=200/101