Tính lim (1/1.3+ 1/3.5+1/5.7+...+1/(2n-1)(2n+1)

Lờigiải

Ta có: 1(2k−1).(2k+1)=1212k−1−12k−1  ,∀k∈ℕ*

Do đó:  11.3+13.5+15.7+...+12n−12n+1

              =121−13+13−15+15−17+...+12n−1−12n+1

              =121−12n+1=n2n+1.

Nên lim11.3+13.5+15.7+...+12n−12n+1

          =limn2n+1=limnn2+1n

          =lim12+1n=12+0=12.

Kết luận: lim11.3+13.5+15.7+...+12n−12n+1=12.