급수 보기 판단

[출제의도] 무한급수의 뜻을 알고 추론하기 ㄱ. $\displaystyle \sum\limits _{n = 1} ^{\infty} \frac{1}{\sqrt{n + 1} + \sqrt{n}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left( \sqrt{n + 1} - 1 \right)$$= \infty$ $\therefore$ $\displaystyle \sum\limits _{n = 1} ^{\infty} a _{n}$은 발산 (참) ㄴ. (반례) $\displaystyle a _{n} = \frac{1}{3 ^{n}} , b _{n} = 2$ (거짓) ㄷ. $a _{1} = 1$, $\displaystyle a _{n + 1} = \frac{1}{n + 1} a _{n}$이므로 $\displaystyle a _{2} = \frac{1}{2} a _{1}$, $\displaystyle a _{3} = \frac{1}{3} a _{2}$, $\cdots$, $\displaystyle a _{n} = \frac{1}{n} a _{n-1}$ $\therefore$ $\displaystyle a _{n} = \frac{1}{2 \times 3 \times 4 \times \cdot \cdot \cdot \times n} a _{1} = \frac{1}{n !}$이므로 $\displaystyle \frac{a _{n+2}}{a _{n}}$$\displaystyle = \frac{1}{n + 1} - \frac{1}{n + 2}$ $\displaystyle \sum\limits _{n=1} ^{\infty} \frac{a _{n+2}}{a _{n}} = \sum\limits _{n=1} ^{\infty} \left( \frac{1}{n + 1} - \frac{1}{n + 2} \right)$$\displaystyle = \frac{1}{2}$ (참) 따라서 옳은 것은 ㄱ, ㄷ