등비수열 부분합과 나머지합

$\displaystyle a _{n} = 1 \cdot \left( \frac{1}{2} \right) ^{n-1} = \left( \frac{1}{2} \right) ^{n-1}$ $\displaystyle \therefore S _{n} = \frac{1 \cdot \left( 1 - \left( \frac{1}{2} \right) ^{n} \right)}{1 - \frac{1}{2}} = 2 - \left( \frac{1}{2} \right) ^{n-1}$, $\displaystyle T _{n} = \frac{\left( \frac{1}{2} \right) ^{n-1}}{1 - \frac{1}{2}} = \left( \frac{1}{2} \right) ^{n-2} = a _{n-1}$ ㄱ) $a _{n} + S _{n} = 2$ ㄴ) $T _{n} = a _{n-1}$ ㄷ) $\displaystyle \lim\limits _{n\rightarrow \infty} {\left( S _{n} + T _{n} \right)} = \lim\limits _{n\rightarrow \infty} {\left( 2 - a _{n} + a _{n-1} \right) = 2}$ $\displaystyle \left( \because \lim\limits _{n\rightarrow \infty {a _{n} = \lim\limits _{n\rightarrow \infty {a _{n-1} = 0 )}}}} \right.$ $\displaystyle \sum\limits _{k=1} ^{\infty} a _{k} = \frac{1}{1 - {\frac{1}{2}}} = 2$ $\therefore$$\displaystyle \lim\limits _{n\rightarrow \infty} \left( S _{n} + T _{n} \right) = \sum\limits _{k=1} ^{\infty} a _{k}$$= 2$ 그러므로 ㄱ, ㄴ, ㄷ 모두 옳다.