Cho dãy số (un) với u(n)=1/1.3+1/3.5+...+1/(2n-1)(2n+1). Khi đó limu(n) bằng?

* Hướng dẫn giải

$\begin{array}{l}{u}_n=\frac{1}{1.3}+\frac{1}{3.5}+\mathrm{...}+\frac{1}{\left(2n-1\right).\left(2n+1\right)}\\ =\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\mathrm{...}+\frac{1}{2n-1}-\frac{1}{2n+1}\right)\\ =\frac{1}{2}.\left(1-\frac{1}{2n+1}\right)\\ \Rightarrow \lim u_n=\lim \frac{1}{2}\left(1-\frac{1}{2n+1}\right)=\frac{1}{2}.\end{array}$