$S=1+\dfrac{1}{2}+\dfrac{1}{4}+...$
$=\Big(\dfrac{1}{2}\Big)^0+\Big(\dfrac{1}{2}\Big)^1+\Big(\dfrac{1}{2}\Big)^2+...$
$=\dfrac{1}{1-\dfrac{1}{2}}$
$=2$
$S=1-\dfrac{1}{3}+\dfrac{1}{9}-\dfrac{1}{27}+...$
$=\Big(\dfrac{-1}{3}\Big)^0+\Big(\dfrac{-1}{3}\Big)^1+\Big(\dfrac{-1}{3}\Big)^2+...$
$=\dfrac{1}{1+\dfrac{1}{3}}$
$=\dfrac{3}{4}$