令m=1
有a(n+1)=a(n)+a(1)+n=a(n)+n+1
a(n)=a(n-1)+n
a(n-1)=a(n-2)+n-1
...
a(2)=a(1)+2
那么把式子左右分别相加,得到a(n)=a(1)+2+...+n=(n+1)n/2
1/a(1)+1/a(2)+...+1/a(2010)
=2/(1*2)+2/(2*3)+...+2/(2010*2011)
=2[1/(1*2)+1/(2*3)+...+1/(2010*2011)]
=2[1/1-1/2+1/2-1/3+...+1/2010-1/2011]
=2[1-1/2011]
=4020/2011