$\sum _{kequals;1}^{\infty}\frac{1}{{2}^{k}}equals;\frac{1}{2}plus;\frac{1}{4}plus;\frac{1}{8}plus;\frac{1}{16}plus;..period;equals;1$
${}$
The sum of the series is 1.
To see this, imagine that we paint a blank canvas in steps. At each step, we paint half of the unpainted area. The total area painted after $n$ steps is therefore the $n$th partial sum, $\sum _{kequals;1}^{n}{\frac{1}{{2}^{k}}}_{}equals;\frac{1}{2}plus;\frac{1}{4}plus;\frac{1}{8}plus;\frac{1}{16}plus;\frac{1}{{2}^{n}}$. The total area remaining unpainted is $\frac{1}{{2}^{n}}$. After an infinite number of steps we will have painted all of the canvas, of which the area is 1.
Click on the canvas to paint one section, or click "Paint All".