거듭제곱근 개수 급수

$n \geq 3$에서 $\displaystyle ( - 3 ) ^{\frac{n - 1}{n}} = ( - 3 ) \cdot \frac{1}{\sqrt[n]{- 3}}$이 실수이므로 $n = 2 k + 1$일 때, $a _{n} = 1$ $n = 2 k + 2$일 때, $a _{n} = 0$(단, $k$는 자연수) $\therefore$$\displaystyle \sum\limits _{n = 3} ^{\infty} \frac{a _{n}}{2 ^{n}} = \frac{a _{3}}{2 ^{3}} + \frac{a _{4}}{2 ^{4}} + \frac{a _{5}}{2 ^{5}} + \frac{a _{6}}{2 ^{6}} + \cdots$ $\displaystyle = \frac{1}{2 ^{3}} + \frac{0}{2 ^{4}} + \frac{1}{2 ^{5}} + \frac{0}{2 ^{6}} + \cdots$ $\displaystyle = \frac{1}{2 ^{3}} + \frac{1}{2 ^{5}} + \frac{1}{2 ^{7}} + \cdots$ $\displaystyle = \frac{\frac{1}{8}}{1 - \frac{1}{4}} = \frac{1}{6}$