You're just adding the elements of the two series pair-wise (which "works" because addition is commutative and associative).

Here are a few additional intermediate steps:

    =  (1 + -1 +  1 + -1 ...)
     + (0 +  1 + -1 +  1 ...)
    
    = (1 + 0) + (-1 + 1) + (1 + -1) + (-1 + 1) ...

Just add 0 in front of the second sum. 0 + x = x, so it is the same.