A simple limit

$\displaystyle S = 1 + -\frac{1}{2} + \frac{1}{4} + -\frac{1}{8} \ldots$

$\displaystyle S = 1 + -\frac{1}{2}(1 + -\frac{1}{2} + \frac{1}{4} \ldots)$

$\displaystyle S = 1 - \frac{S}{2}$

$\displaystyle \frac{3S}{2} = 1$

$\displaystyle \Rightarrow \boxed{ S = \frac{2}{3}}$

$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \frac{1}{32} ...=(1+\frac{1}{4}+\frac{1}{16}+ ...) -(\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+...)=(\frac{1}{1-\frac{1}{4}})-(\frac{\frac{1}{2}}{1-\frac{1}{4}})=\frac{4}{3}-\frac{2}{3}=\Large\color{#3D99F6}{\boxed{\frac{2}{3}}}$

The series can be written as (1 - 1/2) + (1/4 - 1/8) + (1/16 -1/32)... = 1/2 + 1/8 + 1/32... = 1/2 (1 + 1/4 + 1/16...) = 1/2 (1/(1 - 1/4)) = 1/2 (4/3) = 2/3