计算1/(3+根号3）+1/（5倍根号3+3倍根号5）+1/（7倍根号5+5倍根号7）+.+1/（49倍根号47加47倍根号49）

Question

计算1/(3+根号3）+1/（5倍根号3+3倍根号5）+1/（7倍根号5+5倍根号7）+.+1/（49倍根号47加47倍根号49）

Answer 1

显然第n项表达式为1/((2n+1)√(2n-1)+(2n-1)√(2n+1)
分母有理化,得
1/((2n+1)√(2n-1)+(2n-1)√(2n+1))
=((2n+1)√(2n-1)-(2n-1)√(2n+1))/[((2n+1)√(2n-1)+(2n-1)√(2n+1))((2n+1)√(2n-1)-(2n-1)√(2n+1))]
=((2n+1)√(2n-1)-(2n-1)√(2n+1))/[(2n+1)²(2n-1)-(2n-1)²(2n+1)]
=((2n+1)√(2n-1)-(2n-1)√(2n+1))/[(2n+1)(2n-1)*2]
=[(√(2n-1))/(2n-1)-(√(2n+1))/(2n+1)]/2
则
原式=[(1-√3/3)+(√3/3-√5/5)+(√5/5-√7/7)+……+(√47/47-√49/49)]/2
=[1-√49/49]/2
=(1-7/49)/2
=3/7