### Instruction: 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.### Response: I notice that this is a geometric series with a common ratio of $-0.1_2=-\frac{1}{2}_{10}$.
To find the sum of an infinite geometric series, I can use the formula $S=\frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio.
In this case, $a=0.1_2=\frac{1}{2}_{10}$ and $r=-\frac{1}{2}_{10}$.
Plugging these values into the formula, I get $S=\frac{\frac{1}{2}_{10}}{1-\left(-\frac{1}{2}_{10}\right)}=\frac{\frac{1}{2}_{10}}{\frac{3}{2}_{10}}=\frac{1}{3}_{10}$.