sum from n=2 to infinity of 1/(3(2^n))

解

n = 2 \sum \infty \frac{1}{3 ( 2 ^{n} )}

\frac{1}{6}

十進法表記

0.16666 \dots

解答ステップ

n = 2 \sum \infty \frac{1}{3 \cdot 2 ^{n}}

定数乗算の規則を適用する:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= \frac{1}{3} \cdot n = 2 \sum \infty \frac{1}{2 ^{n}}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

= \frac{1}{3} (n = 0 \sum \infty \frac{1}{2 ^{n}} - \frac{1}{2 ^{0}} - \frac{1}{2 ^{1}})

n = 0 \sum \infty \frac{1}{2 ^{n}} = 2

= \frac{1}{3} (2 - \frac{1}{2 ^{0}} - \frac{1}{2 ^{1}})

簡素化

\frac{1}{3} (2 - \frac{1}{2 ^{0}} - \frac{1}{2 ^{1}}) : \frac{1}{6}

= \frac{1}{6}