Why does my weekly income fluctuate wildly when no new services were added?

This most likely is caused by the day/night cycle. During night, commercial income is reduced (except for leisure specialization from the After Dark DLC). The "day" lasts a few weeks, so there will be noticeable swings in budget. Industrial income should also suffer, because the commercial zones do not buy the goods.

This can also be seen under Statistics => City Budget, where the average budget looks like it is modulated by a sinusoidal.

An additional source of variation could be industry from the Industries DLC, where completed production / import can lead to noticeable changes. These should not be periodic, however.