1-1/2+1/3-1/4+...-1/199+1/200(这是分子）1/202^2-2^2+1/203^2-3^2+...+1/300^2-100^2(这是分母）求这个分数的值

Question

1-1/2+1/3-1/4+...-1/199+1/200(这是分子）
1/202^2-2^2+1/203^2-3^2+...+1/300^2-100^2(这是分母）
求这个分数的值

Answer 1

结果应该对,可是有些繁琐
分母：
1/(202^2-2^2)+1/(203^2-3^2)+...+1/(300^2-100^2)
=1/(200*204)+1/(200*206)+.+1/(200*400)
=1/400*(1/102+1/103+1/104+.+1/200)
分子：
1-1/2+1/3-1/4+..+1/99-1/100+1/101-1/102+.+1/199-1/200
=1-1/2+1/3-..+1/102+1/103+.+1/199+1/200-2(1/102+1/104+.+1/200)
=1-1/2+1/3-..+(1/102+1/103+.+1/199+1/200)-(1/51+1/52+.+1/100)
=1-1/2+1/3-...-1/50-2(1/52+1/54+.+1/100)+(1/102+1/103+.+1/199+1/200)
=1-1/2+1/3-...-1/50-(1/26+1/27+.+1/50)+(1/102+1/103+.+1/199+1/200)
=1-1/2+..-1/26+1/27-...+1/49-1/50-(1/26+1/27+.+1/50)+(1/102+1/103+.+1/199+1/200)
=1-1/2+..+1/25-2(1/26+1/28+...+1/50)+(1/102+1/103+.+1/199+1/200)
=1-1/2+..+1/25-(1/13+1/14+...+1/25)+(1/102+1/103+.+1/199+1/200)
=1-1/2+...-1/12-(1/7+1/8+...+1/12)+(1/102+1/103+.+1/199+1/200)
=1-1/2+..-1/6-(1/4+1/5+1/6)+(1/102+1/103+.+1/199+1/200)
=1-1/2+1/3-2*1/4-2*1/6+(1/102+1/103+.+1/199+1/200)
=0+(1/102+1/103+.+1/199+1/200)
=1/102+1/103+.+1/199+1/200
∴分子/分母=1/(1/400)=400