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?

Using techniques that are frequently used by physicists: \begin{align*}
1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots &= \left( 1 - \frac{1}{2} \right) + \left( \frac{1}{4} - \frac{1}{8} \right) + \cdots \\
&= \frac{1}{2} + \frac{1}{8} + \cdots \\
&= \frac{1}{2} \left( 1 + \frac{1}{4} + \frac{1}{16} + \cdots \right) \\
&= \frac{1}{2} \frac{1}{1-\frac{1}{4}} \\
&= \frac{1}{2-\frac{1}{2}} \\
&= \frac{1}{3/2} \\
&= \frac{2}{3}
\end{align*}
Note:  There is no error in the parenthesization in the first line.  We know the series converges (alternating series test), so every subsequence of the partial sums converges to the same thing, so this particular subsequence of the partial sums converges to the same thing.
