Cho dãy số (un) với u(n) = 1-1/2+1/2-1/3+...+1/n-1/n+1 . Khi đó limu(n) bằng?

$u_n=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\mathrm{...}+\frac{1}{n.\left(n+1\right)}$

$=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\mathrm{...}+\frac{n+1-n}{n.\left(n+1\right)}$

$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\mathrm{....}+\frac{1}{n}-\frac{1}{n+1}$

$=1-\frac{1}{n+1}$

$\Rightarrow \lim u_n=\lim \left(1-\frac{1}{n+1}\right)=1.$

Đáp án cần chọn là: C