The series given is:

$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots$

We can see that this is an alternating geometric series. The first term $a$ is 1, and the common ratio $r$ is $-\frac{1}{2}$.

In sigma notation, the series can be written as:

$\sum_{n=0}^{\infty} (-1)^n \left(\frac{1}{2}\right)^n$

This series is an infinite geometric series, which converges when $|r| < 1$.

The sum $S$ of an infinite geometric series is given by the formula:

$S = \frac{a}{1 - r}$

Substituting the values, we have:

$S = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}$

So, the sum of the series is $\frac{2}{3}$.