Hi.
I saw that there were a pair of threads that discussed these series but since it was far away from my knowledge I decided to start my own.
I have just been introduced to these series and found out that the series on the form of:
a_0 + a_1*x + a_2*x^2 + a_3*x^3 + ....

Where
a_0 = 1
a_1 = 0
a_2 = 1
a_3 = 0
a_4 = 1
...
Can be descirbed with the equation (1-x^2)^-1 as long as -1 ≤ x ≤ 1, i.e. it is convergence.

But I also read that even though the answers where x is either smallar than -1 or bigger than 1 seems like nonsens, this isnt the case. For example when x = 2 the author writes

That the equation,(1-x^2)^-1, gives.
Now in which way can -1/3 make any meaning when the series clearly goes to infinity?

There is a concept in complex variable theory called analytic continuation. Using this principal a function analytic within a certain domain (in this case |z|<1) can be extened to the rest of the complex plane, with singularities. Here the singularities are at z=-1 and z=1. Since the power series is equivalent to (1-z^2)^-1 for z within the unit circle, the closed form is the correct analytic continuation.