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.
I know that a geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. In this case, the ratio is $-0.1_2$, which is $-\frac{1}{2}$ in base 10. The first term is $0.1_2$, which is $\frac{1}{2}$ in base 10. So I can plug in $a = \frac{1}{2}$ and $r = -\frac{1}{2}$ into the formula and simplify. I get $S = \frac{1}{2}/(1-(-\frac{1}{2})) = \frac{1}{2}/(\frac{3}{2}) = \frac{1}{3}$. This is the sum of the geometric series as a fraction in base 10. 