求和1/1·3+1/3·5+1/5·7+...+1/(2n+1)(2n-1)
3个回答

1/(2n+1)(2n-1)

=[(2n+1)-(2n-1)]/【2×(2n+1)(2n-1)】

=1/2×[(2n+1)-(2n-1)]/【(2n+1)(2n-1)】

=1/2×[1/(2n-1)-1/(2n+1)

所以1/1*3=1/2×(1-1/3)

1/3*5=1/2×(1/3-1/5)

.

所以原式=1/2×(1-1/3)+1/2*(1/3-1/5)+.+1/2×[1/(2n-1)-1/(2n+1)

=1/2*(1-1/3+1/3-1/5+1/5-1/7+.+1/(2n-1)-1/(2n+1))

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

=1/2*2n/(2n+1)

=n/2n+1