Easy summation question: $S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$ While during physics I encountered a sum I couldn't evaluate:
$$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$
Is there a particular formula for this sum and does it converges?

$$S=1 - \frac {1}{2} + \frac {1}{4} - \frac {1}{8} ...$$ Let this be (1)
Multilply both sides by $\frac {1}{2}$ we get
$$\frac{S}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{8} ...$$ and this be (2)
Now Add (1) and (2)
$$\frac{3}{2} \cdot S=1$$
$$ S=\frac{2}{3}$$
