sum from n=1 to infinity of (10)/(3^n)

解

n = 1 \sum \infty \frac{10}{3 ^{n}}

5

解答ステップ

n = 1 \sum \infty \frac{10}{3 ^{n}}

定数乗算の規則を適用する:

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

= 10 \cdot n = 1 \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}

= 10 (n = 0 \sum \infty \frac{1}{3 ^{n}} - \frac{1}{3 ^{0}})

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

= 10 (\frac{3}{2} - \frac{1}{3 ^{0}})

簡素化

10 (\frac{3}{2} - \frac{1}{3 ^{0}}) : 5

= 5