+537 Find the sum of the base-2 geometric series \(0.1_2-0.01_2+0.001_2-0.0001_2+0.00001_2\ldots\); give your answer as a fraction in which the numerator and denominator are both expressed in base 10.
This is urgent!
+26393 Find the sum of the base-2 geometric series
\(0.1_2-0.01_2+0.001_2-0.0001_2+0.00001_2\ldots\);
give your answer as a fraction in which the numerator and denominator are both expressed in base 10.
\(\begin{array}{|rcll|} \hline 0.1_{2} = 1\cdot 2^{-1} &=& \frac{1}{2} \\ 0.01_{2} = 1\cdot 2^{-2} &=& \frac{1}{4} \\ 0.001_{2} = 1\cdot 2^{-3} &=& \frac{1}{8} \\ 0.0001_{2} = 1\cdot 2^{-4} &=& \frac{1}{16} \\ 0.00001_{2} = 1\cdot 2^{-5} &=& \frac{1}{32} \\ \ldots \\ \hline \end{array}\)
Infinite Geometric Series:
\(\begin{array}{|rcll|} \hline s &=& \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{8} - \dfrac{1}{16} + \dfrac{1}{32} \pm \ldots \quad | \quad a=\frac{1}{2},~ r = -\frac{1}{2} \\\\ s &=& \dfrac{a}{1-r} \quad | \quad |r| < 1. \\\\ s &=& \dfrac{\frac{1}{2}}{1- \left(-\frac{1}{2} \right)} \\\\ s &=& \dfrac{\frac{1}{2}}{1+\frac{1}{2}} \\\\ s &=& \dfrac{\frac{1}{2}}{\frac{3}{2}} \\\\ s &=& \dfrac{1}{2} \cdot \dfrac{2}{3} \\\\ \mathbf{s} & \mathbf{=} & \mathbf{\dfrac{1}{3}} \\ \hline \end{array} \)
+26393 Find the sum of the base-2 geometric series
\(0.1_2-0.01_2+0.001_2-0.0001_2+0.00001_2\ldots\);
give your answer as a fraction in which the numerator and denominator are both expressed in base 10.
\(\begin{array}{|rcll|} \hline 0.1_{2} = 1\cdot 2^{-1} &=& \frac{1}{2} \\ 0.01_{2} = 1\cdot 2^{-2} &=& \frac{1}{4} \\ 0.001_{2} = 1\cdot 2^{-3} &=& \frac{1}{8} \\ 0.0001_{2} = 1\cdot 2^{-4} &=& \frac{1}{16} \\ 0.00001_{2} = 1\cdot 2^{-5} &=& \frac{1}{32} \\ \ldots \\ \hline \end{array}\)
Infinite Geometric Series:
\(\begin{array}{|rcll|} \hline s &=& \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{8} - \dfrac{1}{16} + \dfrac{1}{32} \pm \ldots \quad | \quad a=\frac{1}{2},~ r = -\frac{1}{2} \\\\ s &=& \dfrac{a}{1-r} \quad | \quad |r| < 1. \\\\ s &=& \dfrac{\frac{1}{2}}{1- \left(-\frac{1}{2} \right)} \\\\ s &=& \dfrac{\frac{1}{2}}{1+\frac{1}{2}} \\\\ s &=& \dfrac{\frac{1}{2}}{\frac{3}{2}} \\\\ s &=& \dfrac{1}{2} \cdot \dfrac{2}{3} \\\\ \mathbf{s} & \mathbf{=} & \mathbf{\dfrac{1}{3}} \\ \hline \end{array} \)
+26393 Hello Rom,
i switch it to an alternating series, because the question is: 0.12 - 0.012 + 0.0012 - 0.00012 + 0.000012 +- ...
and the answer is a fraction.
\(\begin{array}{|rcll|} \hline r &=& \dfrac{ {\color{red}-}\dfrac{1}{4} } { \dfrac{1}{2} } = {\color{red}-}\dfrac{1}{2} \\\\ r &=& \dfrac{ \dfrac{1}{8} } { {\color{red}-}\dfrac{1}{4} } = {\color{red}-}\dfrac{1}{2} \\ \ldots \\ \hline \end{array}\)
