sum from k=1 to infinity of 1/(11^k)

Lösung

k = 1 \sum \infty \frac{1}{1 1 ^{k}}

\frac{1}{10}

Dezimale

0.1

Schritte zur Lösung

k = 1 \sum \infty \frac{1}{1 1 ^{k}}

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

= k = 0 \sum \infty \frac{1}{1 1 ^{k}} - \frac{1}{1 1 ^{0}}

k = 0 \sum \infty \frac{1}{1 1 ^{k}} = \frac{11}{10}

= \frac{11}{10} - \frac{1}{1 1 ^{0}}

Vereinfache

\frac{11}{10} - \frac{1}{1 1 ^{0}} : \frac{1}{10}

= \frac{1}{10}

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

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

n = 1 \sum \infty 5 (\frac{3}{7})^{n}

n = 1 \sum \infty \frac{1}{24} (\frac{3}{4})^{n}

n = 0 \sum \infty A n \cdot (- 1)^{n}