证明题2/(x^2-1)+4/(x^2-4)+...+20/(x^2-100)=11/(x-1)(x+10)+11/(x-2)(x+9)+...+11/(x-10)(x+1)2/(x^2-1)+4/(x^2-4)+6/(x^2-9)+...+20/(x^2-100)=11/(x-1)(x+10)+11/(x-2)(x+9)+...+11/(x-10)(x+1)

Question

证明题2/(x^2-1)+4/(x^2-4)+...+20/(x^2-100)=11/(x-1)(x+10)+11/(x-2)(x+9)+...+11/(x-10)(x+1)
2/(x^2-1)+4/(x^2-4)+6/(x^2-9)+...+20/(x^2-100)=11/(x-1)(x+10)+11/(x-2)(x+9)+...+11/(x-10)(x+1)

Answer 1

左边
2/（x^2-1）=1/(x-1)-1/(x+1)
4/（x^2-4）=1/(x-2)-1/(x+2)
.
20/（x^2-20）=1/(x-10)-1/(x+10)
右边
11/（x-1)(x+10）=1/(x-1)-1/(x+10)
.
11/（x-10)(x+1）=1/(x-10)-1/(x+1)
左边右边都＝1/（x-1）+1/（x-2）+1/（x-3）+1/（x-4）+1/（x-5）+1/（x-6）+1/（x-7）+1/（x-8）+1/（x-9）+1/（x-10/）-1/（x+1）-1/（x+2）-1/（x+3）-1/（x+4）-1/（x+5）-1/（x+6）-1/（x+7）-1/（x+8）-1/（x+9）-1/（x+10）