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

Lösung

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

\frac{1}{3}

Dezimale

0.33333 \dots

Schritte zur Lösung

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

Wende die Regel der konstanten Multiplikation:

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

= \frac{2}{3} \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}

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

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

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

Vereinfache

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

= \frac{1}{3}

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