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

解

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

\frac{1}{3}

十進法表記

0.33333 \dots

解答ステップ

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

定数乗算の規則を適用する:

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

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

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

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

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

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

簡素化

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

= \frac{1}{3}