解由1-1/(n+1)²=[(n+1)²-1]/(n+1)²=[n²+2n]/(n+1)²=n(n+2)/(n+1)²
即1-1/2²=1*3/2*2
1-1/3²=2*4/3*3
.
1-1/2009²=2008*2010/2009
1-1/2010²=2009*2011/2010*2010
上述各式相乘
得(1-1/2²)(1-1/3²)(1-1/4²)+.*(1-1/2009²)(1-1/2010²)
=(1*3/2*2)(2*4/3*3)(3*5/4*4).(2008*2010/2009)(2009*2011/2010*2010)
=1/2*2011/2010
=2011/4020