Question

1＋2²／2²－1＋3²／3²－1...＋100²／100²-1

Answer 1

裂项求和的方法

n>2时

n²/(n²-1)

=1+1/[(n-1)(n+1)]

=1+(1/2)*[1/(n-1)-1/(n+1)]

所以,

1＋2²／2²－1＋3²／3²－1...＋100²／100²-1

=1+1+(1/2)[1/1-1/3]+1+(1/2)[1/2-1/4)]+.+1+(1/2)[1/99-1/101]

=100+(1/2)*(1+1/2-1/100-1/101)

=100+(1/2)(3/2-201/10100)

=100+3/4-201/20200