1/15+1/105+1/315+、、、+1/(2n-1)(2n+1)(2n+3)你们谁知道化简结果啊?

题目

1/15+1/105+1/315+、、、+1/(2n-1)(2n+1)(2n+3)你们谁知道化简结果啊?
1/15是十五分之一的意思

答案

设1/(2n-1)(2n+1)(2n+3)=A/(2n-1)+B/(2n+1)+C/(2n+3)
=>A=C=1/8 B=-1/4
=>1/15+1/105+1/315+、、、+1/(2n-1)(2n+1)(2n+3) (n>1)
=[1/1-2/3+1/5+1/3-2/5+1/7+1/5-2/7+1/9+...+1/(2n-1)-2/(2n+1)+1/(2n+3)]/8
={[1+1/3+1/5+...+1/(2n-1)]-2[1/3+1/5+1/7+...+1/(2n+1)]+[1/5+1/7+...+1/(2n+3)]}/8
=[1+1/3-2/3-1/(2n+1)+1/(2n+3)]/8=[1/3-1/(2n+3)(2n+1)]/ (n>1)