lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }=?

lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }=?
lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }=?

lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }=?
1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)
=[2/(1*3)+2/(3*5)+...+2/[(2n-1)(2n+1)]/2
=[(1/1-1/3)+(1/3-1/5)+……+(1/(2n-1)-1/(2n+1)]/2
=[1-1/(2n+1)]/2
=1/2-1/(4n+2)
所以lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }
=lim(x→∞)[1/2-1/(4n+2)=1/2
答案是1/2
1/2{1-1/3+1/3-1/5+...+1/(2n-1)-1/(2n+1)}=n/(2n+1)
lim(x→∞){n/(2n+1)}=1/2

答案是1/2 过程为：
1/2{1-1/3+1/3-1/5+...+1/(2n-1)-1/(2n+1)}=n/(2n+1)
lim(x→∞){n/(2n+1)}=1/2

厉害

1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)
=[2/(1*3)+2/(3*5)+...+2/[(2n-1)(2n+1)]/2
=[(1/1-1/3)+(1/3-1/5)+……+(1/(2n-1)-1/(2n+1)]/2
=[1-1/(2n+1)]/2
=1/2-1/(4n+2)
所以lim(x→∞){ 1/(1*3)+1/(3*5)+...+1/[(2n-1)(2n+1)] }
=lim(x→∞)[1/2-1/(4n+2)=1/2