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

Lösung

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

\frac{3}{7}

Dezimale

0.42857 \dots

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{a ^{c}}{b ^{c}} = (\frac{a}{b})^{c}

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

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

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

n = 0 \sum \infty (\frac{3}{10})^{n} = \frac{10}{7}

= \frac{10}{7} - (\frac{3}{10})^{0}

Vereinfache

\frac{10}{7} - (\frac{3}{10})^{0} : \frac{3}{7}

= \frac{3}{7}

n = 1 \sum \infty e^{- n} + e^{n}

n = 1 \sum \infty \frac{1}{n ! 4 ^{n}}

n = 0 \sum \infty \frac{1}{3 - n}

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

k = 1 \sum \infty k e^{- 7 k}