How do you find the sum of the infinite geometric series 2-2+2-2+..?
The series diverges by typical summation methods, but may converge depending on the method used.
In general, a geometric series will diverge if the common ratio
Still, let's look at this case in a little more detail.
Typically, we refer to
For the series in question, we have
As the sequence
(the fact that the sequence
The particular series mentioned is similar to a famous series known as Grandi's Series. Like Grandi's series, it diverges in conventional summation, but may converge if another method, such as Cesàro summation, is used.
Numberphile has a nice video on the subject here