It is a fact that the *alternating harmonic series*:

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots$$

...converges to $\ln2$. Use this fact to find the sum of the series:

$$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\ldots$$

(Hint: combine each positive term with the one following it.)