You remarked about them stating that the sum of 1-1+1-1+1-1 is equal to a half. This is not a mistake; however it is also not correct. There are two values for this sum, observed by putting in brackets like this: (1-1)+(1-1)... which equals 0, or like this: 1+(-1+1)+(-1+1)=1. As this apparently equals 1 and 0, some mathematicians like to speculate that the answer is in fact 1/2. Without looking at their working, I can't tell whether they approached this answer in the same way, however our lecturer did mention the fact that people do accept the answer of 1/2 when we went through this specific question.