급수 수렴 보기

ㄱ) $\displaystyle \lim\limits _{n \rightarrow \infty} {a _{n} = \frac{1}{2}} \neq 0$이므로 $\displaystyle \sum\limits _{n=1} ^{\infty {a _{n}}}$은 발산한다. ㄴ) $\displaystyle a _{n} = \sum\limits _{k = 1} ^{n} \frac{1}{\sqrt{k + 1} + \sqrt{k}} = \sum\limits _{k=1} ^{n} \left( \sqrt{k + 1} - \sqrt{k} \right) = \sqrt{n + 1} - 1$ $\displaystyle \therefore \lim\limits _{n\rightarrow \infty {a _{n} = \lim\limits _{n\rightarrow \infty {\left( \sqrt{n + 1} - 1 \right) = \infty}}}}$ 이므로 수열 $a _{n}$은 양의 무한대로 발산한다, ㄷ) $\displaystyle \left| 1 + ( a _{1} + a _{2} + \cdots + a _{n _{}} ) ^{{} ^{}} \right| < \frac{1}{n}$ $\displaystyle \leftrightarrow - \frac{1}{n} < 1 + ( a _{1} + a _{2} + \cdots + a _{n _{}} ) ^{{} ^{}} < \frac{1}{n}$ $\displaystyle \rightarrow \lim\limits _{n \rightarrow \infty} {\left( - \frac{1}{n} \right) \leq \lim\limits _{n \rightarrow \infty} {\left\{ 1 + ( a _{1} + a _{2} + \cdots + a _{n _{}} ) ^{{} ^{}} \right\} \leq \lim\limits _{n \rightarrow \infty} {\frac{1}{n}}}}$ $\displaystyle \rightarrow \lim\limits _{n \rightarrow \infty} {\left\{ 1 + ( a _{1} + a _{2} + \cdots + a _{n _{}} ) ^{{} ^{}} \right\}} = 0$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {\left( a _{1} + a _{2} + \cdots + a _{n _{}} \right) = - 1}$ $\displaystyle \therefore \sum\limits _{n=1} ^{\infty} {a _{n}} = \lim\limits _{n\rightarrow \infty {\left( a _{1} + a _{2} + \cdots + a _{n} {) = - 1} \right.}}$