第 057 题 / 100

已知数列 $\left\{ {a}_{n}\right\}$ 的前 $n$ 项和为 ${S}_{n},{a}_{1} = \frac{1}{2}$ , 且 $\frac{{a}_{n}}{{a}_{n} + 1} = 2{a}_{n + 1}$ ,记 $\left\{ \frac{1}{{a}_{n}}\right\}$ 的前 $n$ 项和为 ${T}_{n}$ ,则( )
A. ${a}_{2} = \frac{1}{6}$
B. $\left\{ \frac{1}{{a}_{n}}\right\}$ 是等比数列
C. ${T}_{n} = {2}^{n + 2} - {2n} - 4$
D. $\frac{1}{2} - \frac{1}{{2}^{n + 1}} < {S}_{n} \leq 1 - \frac{1}{{2}^{n}}$