How do you find the sum of the infinite geometric series 2-2+2-2+..?

1 Answer
Dec 20, 2015

The series diverges by typical summation methods, but may converge depending on the method used.

Explanation:

In general, a geometric series will diverge if the common ratio #r# between terms does not have #|r| < 1#. In this case, #r = -1#, meaning #|r| = 1# and so the series should diverge.

Still, let's look at this case in a little more detail.

A series #sum_(k=0)^ooa_k# may be defined as #lim_(n->oo)sum_(k=0)^na_k#.

Typically, we refer to #sum_(k=0)^na_k# as the #n^(th)# partial sum of the series (often denoted #S_n#).

For the series in question, we have

#S_n = sum_(k=0)^n(-1)^k*2 = {(2" if n is even"),(0" if n is odd"):}#

As the sequence #(S_n)# simply alternates between #2# and #0#, it should be clear that #sum_(k=0)^oo(-1)^k*2= lim_(n->oo)S_n# does not converge to any value
(the fact that the sequence #(S_n)# does not converge may be shown simply using an #epsilon-delta# proof).


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