Here's how to write $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - ....$$ using the $\sum$ notation: $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - .... = \sum_{i=0}^\infty \frac{1}{2i+1}- \frac{1}{4i+2}-\frac{1}{4i+4}.$$
Here's how to write $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - ....$$ using the $\sum$ notation: $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - .... = \sum_{i=0}^\infty \frac{1}{2i+1}- \frac{1}{4i+2}-\frac{1}{4i+4}.$$