k에 따른 수열의 극한

$\displaystyle a _{k} = \lim\limits _{n \rightarrow \infty} {\frac{\left( \frac{6}{k} \right) ^{n + 1}}{\left( \frac{6}{k} \right) ^{n} + 1} ( k = 1 , 2 , 3 , \cdots )}$ 에서 (ⅰ) $k < 6$ 일 때, $\displaystyle \frac{6}{k} > 1$이므로, 분모, 분자를 $\displaystyle \left( \frac{6}{k} \right) ^{n}$으로 나누면 $\displaystyle a _{k} = \lim\limits _{n \rightarrow \infty} {\frac{\left( \frac{6}{k} \right)}{1 + \left( \frac{k}{6} \right) ^{n}} = \frac{6}{k} ( \because 0 < \frac{k}{6} < 1 )}$ (ⅱ) $k = 6$일 때, $\displaystyle a _{k} = \frac{1}{1 + 1} = \frac{1}{2}$ (ⅲ) $k > 6$ 일 때, $\displaystyle 0 < \frac{6}{k} < 1$ 이므로 $\displaystyle a _{k} = \frac{0}{0 + 1} = 0$ $\displaystyle \therefore \sum\limits _{k = 1} ^{10} ka _{k} = \sum\limits _{k = 1} ^{5} k \times \frac{6}{k} + \sum\limits _{k = 6} ^{6} k \times \frac{1}{2} + \sum\limits _{k = 7} ^{10} k \times 0$ $\displaystyle = 6 \times 5 + 6 \times \frac{1}{2} = 33$