[출제의도] 무한급수의 합이 수렴을 묻는 문제로 무한급수의 합의 수렴성의 이해를 측정하는 문제이다.
ㄱ. a n = n − 1 2 n + 1 \displaystyle a _{n} = \frac{n - 1}{2 n + 1} a n = 2 n + 1 n − 1 lim n → ∞ a n = 1 2 ≠ 0 \displaystyle \lim\limits _{n \rightarrow \infty} {a _{n}} = \frac{1}{2} \ne 0 n → ∞ lim a n = 2 1 = 0 ∑ n = 1 ∞ n − 1 2 n + 1 \displaystyle \sum\limits _{n=1} ^{\infty} \frac{n - 1}{2 n + 1} n = 1 ∑ ∞ 2 n + 1 n − 1 r = − 1 2 \displaystyle r = - \frac{1}{\sqrt{2}} r = − 2 1 − 1 < r < 1 - 1 < r < 1 − 1 < r < 1 ∑ n = 1 ∞ ( − 1 2 ) n \displaystyle \sum\limits _{n=1} ^{\infty} \left( - \frac{1}{\sqrt{2}} \right) ^{n} n = 1 ∑ ∞ ( − 2 1 ) n ∑ n = 1 ∞ 1 n + 1 + n = lim n → ∞ ∑ k = 1 n 1 k + 1 + k \displaystyle \sum\limits _{n=1} ^{\infty} \frac{1}{\sqrt{n + 1} + \sqrt{n}} = \lim\limits _{n \rightarrow \infty} {\sum\limits _{k=1} ^{n}} \frac{1}{\sqrt{k + 1} + \sqrt{k}} n = 1 ∑ ∞ n + 1 + n 1 = n → ∞ lim k = 1 ∑ n k + 1 + k 1 = lim n → ∞ ∑ k = 1 n ( k + 1 − k ( k + 1 + k ) ( k + 1 − k ) ) \displaystyle = \lim\limits _{n \rightarrow \infty} {\sum\limits _{k=1} ^{n}} {\left( \frac{\sqrt{k + 1} - \sqrt{k}}{( {\sqrt{k + 1} + \sqrt{k} ) ( \sqrt{k + 1} - \sqrt{k} )}} \right)} = n → ∞ lim k = 1 ∑ n ( ( k + 1 + k ) ( k + 1 − k ) k + 1 − k ) = lim n → ∞ ∑ k = 1 n ( k + 1 − k ) \displaystyle = \lim\limits _{n \rightarrow \infty} {\sum\limits _{k=1} ^{n}} {( \sqrt{k + 1} - \sqrt{k} )} = n → ∞ lim k = 1 ∑ n ( k + 1 − k ) = lim n → ∞ ( n + 1 − 1 ) \displaystyle = \lim\limits _{n \rightarrow \infty} {( \sqrt{n + 1} - 1 )} = n → ∞ lim ( n + 1 − 1 )